Everton moved out of the bottom three after goals from Bernard and Gylfi Sigurdsson gave them a 2-0 win over West Ham to relieve the pressure on Marco Silva.
Toffees boss Silva labelled the visit of Manuel Pellegrini's side as a "must-win" game - and his players responded as confidence flowed following Bernard's 17th-minute opener from a tight angle.
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Everton were good value for their victory, hitting the woodwork twice, but they had to wait until stoppage time to put the seal on a much-improved performance through substitute Sigurdsson's superb strike (90+2).
West Ham missed the chance to move up to third temporarily after a second straight loss, and after the later results, dropped to ninth - only two points ahead of Everton in 13th.
Everton: Pickford (7), Sidibe (7), Keane (7), Mina (7), Digne (7), Davies (8), Gomes (9), Walcott (8), Iwobi (8), Bernard (7), Richarlison (7).
Subs: Holgate (n/a), Sigurdsson (7), Kean (n/a).
West Ham: Jimenez (8), Fredericks (6), Diop (7), Ogbonna (7), Masuaku (6), Noble (5), Rice (5), Anderson (5), Fornals (5), Lanzini (6), Haller (6).
Subs: Yarmolenko (7), Wilshere (6), Ajeti (6).
Man of the match: Andre Gomes.
How Everton dominated the Hammers
The Irons went into the game unbeaten away from home as part of their best Premier League start in four seasons, but, after Manuel Lanzini saw an early free-kick fizz wide of Jordan Pickford's post, it was Everton who provided a better demonstration of their flickering top-six aspirations.
"Pressure is normal in football," Silva said after this pivotal victory. "When you are in a very good moment, the pressure is there to win more games. When you are not - and the last results were not good enough - we have to react. The pressure is always there."
The Everton boss made a big decision in opting to drop Sigurdsson as part of five changes, and it was one of his alterations who sparked the reaction.
Theo Walcott appeared to have lost his way at Goodison, but this was one of his best games in an Everton shirt - and it was the winger's cute pass that found Bernard, who cut inside and scored from a tight angle.
- Marco Silva made five changes against West Ham as Djibril Sidibe was handed a first start in place of the suspended Seamus Coleman while Tom Davies, Theo Walcott, Andre Gomes and Bernard came in.
- Manuel Pellegrini made a change at left-back as Arthur Masuaku took the place of Aaron Cresswell. Pablo Fornals started as Andriy Yarmolenko dropped to the bench in two changes from the side that was beaten 2-1 by Crystal Palace.
Roberto was at fault after being beaten at his near post, and the Spaniard, deputising in the West Ham goal for the injured Lukasz Fabianski, was very nearly beaten cheaply again five minutes later when Richarlison's shot came back off the woodwork.
The visitors could barely hold a candle to Everton's greater intensity and quality during the opening 45 minutes and Pellegrini introduced Andriy Yarmolenko at the interval. The Ukrainian - who scored twice in a 3-1 win in this fixture last season - carved out an immediate chance but Pablo Fornals dragged his shot wide.
Everton responded to that warning sign as they regained the initiative after Walcott saw his venomous drive cannon off the crossbar just after the hour-mark. Silva must have feared it would turn into another one of those days when Alex Iwobi somehow failed to convert when put through on goal, with the forward only managing to fire straight at Roberto.
Moments later, referee Paul Tierney saw an infringement from the impressive Andre Gomes for a block as Yerry Mina rose to head into the net, and Tierney then overlooked a foul from Arthur Masuaku on Djibril Sidibe as the resulting corner led to Angelo Ogbonna's instinctive shot being kept out brilliantly by Pickford.
But Sigurdsson would come on for the final few minutes to underline his undoubted quality as he collected Richarlison's pass to cut inside and unleash an unstoppable trademark strike that flew beyond Roberto to secure three points that have never felt more welcome to Silva.
What the managers said
Everton's Marco Silva: "We showed everything that I asked and the fans showed their support. The quality of our game was very good. We always created space and created moments in the game to win it. We clearly deserved the three points and the collective performance was very good.
"We needed the crowd behind the team and they did more than that. The players who came in took their opportunity and there were some individual performances of a very high level."
West Ham's Manuel Pellegrini: "I'm disappointed because I think we didn't have a good performance. We knew before the game that it was a very tough game in a difficult stadium and the position of Everton reflected how much they needed these points. They ran and they fought for the 95 minutes and they deserved the win.
"If I remember the last game we played here, we came in the same position as Everton were before the game, losing four games in a row, so that is something that allowed them to demand more from their players and that's what they did."
Analysis: Everton find solution to set-piece problem
Sky Sports' Ben Grounds:
Silva called on his players to "embrace the moment" as they looked to end a four-game losing run, and 19 attempts on goal reflected the emphatic response of his players on a day when the result was everything.
The Portuguese's team selection reflected a manager looking for answers. The £45m record signing was left out, while he started without a recognised striker. Pressure breeds panic, but this was a performance that demonstrated that he still retains the trust of his players.
It has been well-documented that 46 per cent of the 13 goals that have been conceded this term have come from set-pieces, but, in keeping a clean sheet for the first time in two months, the belief will now be that marginal gains will come from addressing this clear area of weakness.
West Ham were second best throughout this contest, but they very nearly snatched an equaliser when Ogbonna failed to hook in Sebastien Haller's knockdown from one of only two corners Everton conceded all afternoon.
Cutting out the concession of set-pieces is certainly one way of solving the problem.
Man of the match: Andre Gomes
It was back in March when Everton produced arguably their best performance under Silva at the London Stadium. Gomes was instrumental that day, controlling the midfield with Mark Noble absent for West Ham due to a family bereavement that week.
The Portuguese is a fans' favourite on Merseyside, but his performances are often dependent on the outcome of his first ball.
The former Barcelona midfielder is a confidence player, but on a day when Everton simply could not afford anything but a win, Gomes stood up to the challenge and replicated that complete performance against the Hammers some seven months ago.
Thirty-two of Gomes' total of 45 passes were into the opposition half, providing his side with purpose to their play. Furthermore, no player gained possession on more occasions than Gomes' 15, allowing Everton's fluid front four to wreak havoc.
- Everton's last six Premier League wins have all come at Goodison Park; the Toffees have won seven of their last nine home league matches (W7 D0 L2).
- West Ham suffered their first away Premier League defeat of the season (W1 D3 L1), leaving only Liverpool and Sheffield United unbeaten away from home this season.
- Everton's Theo Walcott has been involved in 10 Premier League goals against West Ham (6 goals, 4 assists) - against no side has he been involved in more (also 10 against Newcastle and Tottenham).
Everton travel to face Brighton at the Amex Stadium next Saturday; kick-off is at 3pm. West Ham host Sheffield United at the same time at the London Stadium.