Bournemouth manager Eddie Howe says his side's dominant victory over Brighton proves they are good enough to survive in the Premier League.
While the Cherries remain in the bottom three, victory lifts them up a place to 18th and eases some of the pressure on the 42-year-old, while Brighton are two points ahead in 15th.
A Harry Wilson strike, Pascal Gross' own-goal and Callum Wilson's first goal for 16 league games ended Bournemouth's six-game winless run, with Aaron Mooy scoring late on for the Seagulls.
"It feels big at the moment, but time will tell," Howe said. "We knew how big tonight was in so many different ways.
"There's a lot of clubs up and down the country that maybe wouldn't have been given the time and support that we have, and we have to recognise that and show our appreciation.
"It's not that the team's not good enough - we are good enough, we just need that feeling that we can play the football that we want to, freely, and with total belief and no doubts.
"We're desperate to retain our Premier League status and stay here - I love the division, I love everything about it, and I know how important it is for the club."
Bournemouth needed to show people we are not finished - Howe
This was a welcome return to winning ways for Bournemouth who arrived at this fixture having taken just four points from the previous 36 available to them.
The Cherries also possessed the worst home record in the top flight prior to kick-off, something that seemed unimaginable back on 2 November, when they last won at home by beating Manchester United to move up to seventh in the table.
And, as Brighton took charge in the first half hour, it looked like another game was passing them by until a superb save from home keeper Aaron Ramsdale to deny Neal Maupay, and an equally important Nathan Ake block on Maupay's follow up, seemed to turn the contest around.
The hosts, who had not had an effort on goal of any description until then, suddenly sparked into life as Callum Wilson's influence grew.
Howe's side created three chances in quick succession before Harry Wilson kept his cool to dispatch a low drive past Brighton goalkeeper Mat Ryan from 12 yards.
Callum Wilson's jostling presence contributed to their second, with the ball flicking off his marker Gross from a Diego Rico corner.
The forward then rounded Ryan for the third - calmly rolling the ball in from close range - to show that he has not lost his composure in front of goal during a period in which Bournemouth have averaged less than three shots on target per game.
Brighton undermined by lack of firepower
Graham Potter labels loss a 'really disappointing' evening
In many ways this performance typified Brighton's campaign so far.
Graham Potter's side looked fluent in possession in both halves but were missing the thrust in attack to make the most of their dominance.
While they had 21 shots on goal Mooy's superb left-foot strike into the top right corner was scant consolation.
"We played really well for the first 30 minutes," said Brighton boss Graham Potter.
"They were struggling to get out of their half. We pushed and pushed but in the end it's a disappointing evening. The result is the most important thing."
Alireza Jahanbakhsh and Bernardo had speculative efforts early on, but Maupay will feel he could have scored from either of his two close-range efforts in quick succession.
The 23-year-old Frenchman carried a weight of great expectation early in his career after becoming the second-youngest scorer in Ligue 1 history at 16, but this proved another difficult Premier League occasion for him.
Maupay, who has not scored since the 1-1 draw at Crystal Palace on 16 December, had just 34 touches of the ball and at times appeared isolated.
He does, though, remain the Seagulls' top scorer with seven goals, and with Aaron Connolly, introduced as part of a triple substitution on the hour mark, next on the list with three alongside Leandro Trossard and defender Adam Webster, it suggests Brighton may need to add to their forward line.
Man of the match - Aaron Ramsdale (Bournemouth)
Aaron Ramsdale impressed with seven saves for Bournemouth to lay the platform for their win
Callum Wilson's barren run - the stats
Bournemouth have won five of their last six home league meetings with Brighton (D1), winning all three of their Premier League games against the Seagulls at the Vitality Stadium.
Bournemouth have won their first Premier League home game since November 2019 (1-0 against Man Utd), ending a run of five such games without a victory (D1 L4) and scoring more goals today (3) than in those previous five games combined (2).
Brighton have lost eight of their last 11 Premier League away games (W1 D2), conceding two or more goals in seven of those games
Bournemouth ended a run of 450 minutes without a goal in the Premier League, since scoring in their 1-1 draw with Arsenal in December 2019.
Bournemouth's Harry Wilson is the club's top goalscorer this season in the Premier League with seven goals, averaging a goal every 173 minutes - the best such ratio among Premier League midfielders this season (min 2 goals scored).
Callum Wilson has ended a run of 1,319 minutes without a Premier League goal for Bournemouth, since netting against West Ham in September 2019.
Brighton's Aaron Mooy has scored twice in his last four Premier League appearances, with both goals coming against Bournemouth. He had managed just one goal in his previous 30 such games.
What's next?
Bournemouth host Arsenal in the FA Cup fourth round on Monday 27 January (20:00 GMT). The Cherries' next Premier League game is at home to Aston Villa on Saturday, 1 February (15:00 GMT).
Brighton are next in action when they travel to London Stadium to face West Ham in the Premier League on Saturday, 1 February (15:00 GMT).
Newcastle boss Steve Bruce said his side's comeback at Everton was "remarkable" as two injury-time goals from substitute Florian Lejeune earned a dramatic point.
Bruce's injury-hit side were slipping towards defeat after Moise Kean scored his first Toffees goal before Dominic Calvert-Lewin made it 2-0 to the hosts with his 12th of the season.
But as the clocked ticked into injury time, Lejeune took advantage of some slack defending at a corner to crash an overhead kick past Jordan Pickford in the 94th minute.
And a minute later, Carlo Ancelotti's side threw away what was an otherwise good performance by failing to clear a free-kick with French defender Lejeune taking advantage of another scramble to fire in.
It was a sensational end to a game which Everton had dominated, as Bruce's side added to the injury-time win they earned against Chelsea on Saturday, sending their travelling supporters into delirium.
Bruce, whose side are seven points clear of the relegation zone, said: "It's quite remarkable. I can't fault the players for their endeavour or effort.
"We probably didn't deserve to get anything but we didn't give up. It just shows you what can happen.
"It's a wacky game that's for sure - but it's great. It just shows you shouldn't leave a game early."
Everton will be dismayed by how they let the game slip from their grasp after Kean produced his best performance of the season before being substituted to a standing ovation on 70 minutes.
The Toffees could have narrowed the gap to fifth-placed Manchester United to two points but are now 12th, one place above Newcastle who also have 30 points.
Ancelotti said: "I'm really pleased for the way we played. The players are really sad at this moment, but I told them I have more experience than them.
"I've lost a Champions League final after leading 3-0 [with AC Milan against Liverpool] so it can happen sometimes."
Everton throw away positive performance
Everton played a 'fantastic' game - Carlo Ancelotti
For 93 minutes, Everton looked like they were in control and had answered calls from Ancelotti to be braver, while Kean emphatically answered his manager's hopes for a quick improvement.
Having failed to score in his previous 21 appearances, the 19-year-old was a menace alongside Calvert-Lewin, who also repaid Ancelotti's faith in him after the Italian tipped the 22-year-old for an England place.
Kean has been through a difficult spell since he arrived on Merseyside for £25m, having been disciplined for lateness and suffering the humiliation of being a substituted substitute during Duncan Ferguson's brief reign.
And it looked like his lean spell might continue when he somehow failed to connect with Theo Walcott's low cross before his moment of huge relief arrived.
Gathering Bernard's through ball, he took a poor touch, but after steadying himself, he fired under Martin Dubravka before running away to the corner flag in delight where he was greeted by Everton's thankful supporters.
There were further chances to follow for the Italian international, but Calvert-Lewin showed his budding promise once again by making it 2-0 after collecting Lucas Digne's pass and clipping in left-footed.
When Kean left the field, he was cheered by all four stands at Goodison Park and probably felt he had done enough to earn a third successive home win for Everton.
Newcastle had other ideas, but Ancelotti said of Kean: "The goal was good for his confidence, of course, and after that he played really well.
"He worked hard, he pressed forward, what I asked, he did. He has to improve, because he's young, he has to improve technically and tactically, but he's on the right way."
Newcastle profit from Everton's panic
Steve Bruce praises Magpies' fighting spirit in 'remarkable' game
In keeping with their injury-time win over Chelsea on Saturday, Newcastle spent most of the match defending, but their lack of possession had little effect on the result.
Whereas Chelsea paid for their poor finishing, Everton were far more clinical and in truth could have had several other goals as Bernard and Kean wasted good chances.
But perhaps the injury crisis which has engulfed Newcastle is galvanising Bruce's side.
With nine players on the treatment table, which was added to by a season-ending injury to Jetro Willems on Saturday, Bruce had to play two midfielders in wing-back positions and for much of the game they were disjointed.
They had trouble containing Kean and the impressive Bernard down Everton's left.
However, once the pair were withdrawn, Everton began to retreat.
But having failed to register a shot on target until Lejeune's late double there was little sign of what was to come in injury time.
Newcastle merrily profited from two moments of panic in the Everton box after a performance from the hosts which means Ancelotti has lost only one of six league games in charge.
Man of the match - Moise Kean (Everton)
The result may not have gone Everton's way but Kean was rewarded for his efforts up front with a maiden goal for the Toffees
'We keep showing determination' - what they said
Newcastle manager Steve Bruce speaking to Match of the Day: "It's quite remarkable. I can't fault the players for their endeavour. Joelinton was struggling after 50 minutes; I nearly put a centre-half up front.
"We keep showing determination. We were a tired team tonight but we found a bit of energy from somewhere. When we scored Mike Dean said there are 40 seconds left - and you're thinking 'can we score in 40 seconds?' We rode our luck certainly but they never gave up.
"There is a good spirit and attitude in this team - we looked tired but we played people out of position. We didn't have much coming off the bench in terms of forward players but the attitude was fantastic.
"It's a wacky game that's for sure - but it's great. It just shows you shouldn't leave a game early."
Everton manager Carlo Ancelotti speaking to Match of the Day: "There are things in football you cannot control. We conceded without any reason but the performance was good.
"We played a fantastic game and we were unlucky but nothing changes - our Premier League continues, we have to stay focused. We were two-nil up, had opportunities to score and we played attacking football. I'm really pleased with the performance. These results are not frequent but they can happen.
"We have to stay on the game for 90 minutes but I'm not saying anything to my players. They played well and it can happen. It's just unlucky for this game."
Magpies celebrate late goals - the stats
There were just 102 seconds between Newcastle's two goals against Everton. It is also the first time in Premier League history that the Magpies have scored two goals in the 90th minute onwards in a single game.
Everton have conceded seven goals in the 90th minute onwards in the Premier League this season, four more than any other side in the division.
Newcastle defenders have scored 12 goals in the Premier League this season, four more than any other side in the competition.
Seven of Lucas Digne's nine Premier League assists have come at Goodison Park, including each of his five this season.
Dominic Calvert-Lewin is the first Englishman to score 10 or more goals in a Premier League season for Everton since Wayne Rooney in 2017-18.
At 22 years and 311 days old, Calvert-Lewin is the third-youngest Everton player to reach 100 Premier League appearances for the club after Michael Ball (21y 68d) and Ross Barkley (22y 60d).
Everton's Bernard has been directly involved in four goals in his 15 Premier League appearances this season (2 goals, 2 assists), as many as he recorded in 34 games in his debut campaign last season (1 goal, 3 assists).
Moise Kean has scored his first goal for Everton with what was his 26th shot in all competitions. He's the first Italian to score a Premier League goal for the Toffees since Alessandro Pistone in April 2002. Furthermore, he is the first Italian to score under Carlo Ancelotti in the Premier League.
What's next?
Newcastle host Oxford United on Saturday in the FA Cup fourth round (15:00 GMT), while Everton are next in action on Saturday, 1 February in the Premier League against Watford at Vicarage Road (15:00).
