Dominic Calvert-Lewin's first-half winner boosted Everton's European hopes and dealt West Ham's Champions League aspirations a huge blow in the process, as the Toffees earned a hard-fought victory at the London Stadium.
With 24 minutes on the clock, Calvert-Lewin grabbed his 16th goal of the season after making a clever run to latch onto Ben Godfrey's through-ball and finish past Lukasz Fabianski with the visitors' first shot on target.
With the momentum behind them, the Toffees could have been two ahead when Gylfi Sigurdsson's free-kick forced a fine save from the West Ham goalkeeper. And Fabianski was needed once more to deny Richarlison's angled shot.
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West Ham should have been level before the break however when Pablo Fornals' deep cross picked out the unmarked Said Benrahma but he inexplicably missed the target from point-blank range and the wait for his first goal for the club goes on.
There was more frustration for David Moyes' side after the restart as Vladimir Coufal hit the post and with the goal gaping substitute Jarrod Bowen failed to convert from the rebound.
The hosts continued to pour numbers forward without registering a shot on target and they were almost caught out defensively as Everton missed the opportunity to seal the game late on, with substitute Josh King heading a cross onto the crossbar.
Defeat means West Ham remain five points behind Leicester in fourth with just three games remaining, while Everton close the gap on sixth-placed Liverpool to just two points.
TALKING POINT - Everton inflict damaging defeat on the Hammers
Brendan Rodgers will breathe a huge sigh of relief this evening. Having watched his Leicester side fall to a shock 4-2 home defeat to Newcastle on Friday night, he would have watched this encounter from behind the sofa, knowing West Ham had a huge opportunity to move to within touching distance of their Champions League spot. But, in truth, in the biggest game of their campaign, the Hammers never played to their potential. The result almost certainly ends their Champions League dream; they must win their three remaining games and even that is unlikely to be enough.
Huge credit must go to the visitors, of course. Everton had won four of their five games in London before today, and with aspirations of their own to play in Europe next season, they were full of confidence, and after taking the lead, dug really deep to take all three points to claim a remarkable 11th win on the road this season.
So while this is a damaging defeat for the Hammers, it's a massive three points for Everton.
MAN OF THE MATCH - Dominic Calvert-Lewin (Everton)
Scored the only goal - but there was much more to his game. He worked tirelessly up front, winning flick-ons, linking play and setting up chances for his team mates. Deserved to be on the winning side.
West Ham: Fabianski 7, Cresswell 5, Coufal 6, Diop 5, Dawson 5, Fornals 6, Soucek 6, Benrahma 4, Lanzini 5, Lingard 5, Antonio 6.. subs: Yarmolenko 5, Bowen 5, Fredericks 5.
Everton: Pickford 6, Coleman 6, Digne 6, Keane 7, Mina 7, Godfrey 8, Allan 8, Davies 7, Sigurdsson 7, Richarlison 7, Calvert-Lewin 9*.. subs: Delph N/A, King N/A.
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24' - GOAL! West Ham 0-1 Everton (Dominic Calvert-Lewin): The visitors strike a huge blow to West Ham's Champions League dreams! Calvert-Lewin makes a clever run to get the wrong side of Dawson and he's picked out beautifully by Godfrey, sliding the ball past Fabianski.
39' - Close! West Ham should be level! Fornals' cross to the far post is inch-perfect for the unmarked Benrahma but he gets his header all wrong and it's a real let off for Everton!
61' - Off the post! How have West Ham not scored?! Coufal's first-time angled shot cannons off the woodwork and the rebound falls kindly for Bowen, but with the goal gaping he fails to make the right connection and Everton survive!
84' - Woodwork! King, with his first touches, sets up an attacking move and then heads Calvert-Lewin's cross off the post! West Ham are still alive in this contest, just!
- Only Manchester City (13) have won more Premier League away games than Everton (11) this term, with the Toffees’ 11 away victories being their most in a top-flight campaign since 1984-85 (12); a season in which they won the title.
- Dominic Calvert-Lewin has now scored 16 Premier League goals this season, the joint-most by an Englishman for the Toffees alongside Tony Cottee's 16 back in 1993-94.
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