Match ends, Everton 1, West Ham United 3.
West Ham halted their disastrous start to the season with a first Premier League win under Manuel Pellegrini as they outplayed Everton in a lively contest at Goodison Park.
Andriy Yarmolenko's first Hammers goal opened the scoring, the Ukrainian finishing off a slick counter that went from the halfway line to the net in three passes, Marko Arnautovic with the assist.
Yarmolenko got his second in the 31st minute with a fine curled effort from just inside the box after a Jordan Pickford error gave possession to Mark Noble.
Just before the break, Gylfi Sigurdsson powered in a header from Jonjoe Kenny's cross, converting after Cenk Tosun missed two good chances.
West Ham were coming in on the back of four consecutive defeats - a fifth straight loss would have constituted their worst ever opening of a league campaign.
But Arnautovic ensured that run came to an end when he struck clinically after a one-two with Pedro Obiang, the Hammers again profiting on the break despite Everton's spirited start to the second half.
The Toffees were unbeaten under new manager Marco Silva before Sunday's match, but had only won once, and they will regret the handful of individual errors and missed chances that was ultimately the difference between the two sides.
Even in the game's latter stages they were pouring forward but could still not find the net, substitute Oumar Niasse rattling the crossbar from close range after Lucas Digne's cross.
Victory lifts the Hammers up to 16th, with Burnley taking their place at the bottom of the table after losing 1-0 at Wolves earlier in the day. Everton are now 10th, with six points from their first five games.
West Ham were booed off after their previous game, a 1-0 home defeat by Wolves. Ex-England captain Alan Shearer described that showing as "embarrassing, hopeless and pathetic".
This was a performance much closer to what fans will have been expecting since a summer spend of £100m and the arrival of ex-Real Madrid manager Pellegrini, title winner with Manchester City in 2014.
They were far better organised at the back, more confident in possession, and played with real energy against a full-blooded Everton side who were just determined to attack.
Lukas Fabianksi's goal withstood seven first-half attempts before Sigurdsson's unstoppable bullet header, and on the break Felipe Anderson was exceptional in his link-up play with Yarmolenko and Arnautovic, who was forced off in the second half.
Pellegrini said before this fixture he "trusts now more than ever" his ability to achieve at the highest level. His players backed him up with an impressive counter-attacking performance. Games against Chelsea and Manchester United are up next.
More to follow.
Match ends, Everton 1, West Ham United 3.
Second Half ends, Everton 1, West Ham United 3.
Foul by Jonjoe Kenny (Everton).
Michail Antonio (West Ham United) wins a free kick on the left wing.
Bernard (Everton) is shown the yellow card for a bad foul.
Foul by Bernard (Everton).
Robert Snodgrass (West Ham United) wins a free kick on the right wing.
Bernard (Everton) wins a free kick in the attacking half.
Foul by Carlos Sánchez (West Ham United).
Foul by Lucas Digne (Everton).
Robert Snodgrass (West Ham United) wins a free kick on the right wing.
Attempt missed. Oumar Niasse (Everton) header from the centre of the box is too high. Assisted by Lucas Digne with a cross following a set piece situation.
Robert Snodgrass (West Ham United) is shown the yellow card for a bad foul.
Bernard (Everton) wins a free kick on the left wing.
Foul by Robert Snodgrass (West Ham United).
Attempt blocked. Michail Antonio (West Ham United) right footed shot from the centre of the box is blocked. Assisted by Felipe Anderson.
Lucas Digne (Everton) wins a free kick in the defensive half.
Foul by Michail Antonio (West Ham United).
Corner, West Ham United. Conceded by Lucas Digne.
Foul by Idrissa Gueye (Everton).
Robert Snodgrass (West Ham United) wins a free kick on the right wing.
Oumar Niasse (Everton) hits the bar with a right footed shot from the centre of the box. Assisted by Lucas Digne with a cross.
Substitution, West Ham United. Robert Snodgrass replaces Andriy Yarmolenko.
Foul by Lucas Digne (Everton).
Andriy Yarmolenko (West Ham United) wins a free kick on the right wing.
Offside, Everton. Jonjoe Kenny tries a through ball, but Oumar Niasse is caught offside.
Corner, West Ham United. Conceded by Jonjoe Kenny.
Corner, Everton. Conceded by Carlos Sánchez.
Attempt blocked. Carlos Sánchez (West Ham United) left footed shot from outside the box is blocked.
Attempt blocked. Michail Antonio (West Ham United) right footed shot from the left side of the box is blocked. Assisted by Arthur Masuaku.
Substitution, Everton. Ademola Lookman replaces Theo Walcott.
Delay over. They are ready to continue.
Delay in match Andriy Yarmolenko (West Ham United) because of an injury.
Attempt blocked. Gylfi Sigurdsson (Everton) right footed shot from outside the box is blocked.
Declan Rice (West Ham United) is shown the yellow card for a bad foul.
Theo Walcott (Everton) wins a free kick in the attacking half.
Foul by Declan Rice (West Ham United).
Attempt missed. Felipe Anderson (West Ham United) right footed shot from outside the box misses to the right. Assisted by Arthur Masuaku.
Substitution, West Ham United. Carlos Sánchez replaces Mark Noble.
Substitution, Everton. Oumar Niasse replaces Cenk Tosun.