Michail Antonio, Diafra Sakho and Dimitri Payet scored inside 12 minutes to ruin Everton's lead and claim a 3-2 victory in Saturday afternoon's Premier League clash at Goodison Park.
Romelu Lukaku scored his eighth goal in as many appearances against the Hammers, while Aaron Lennon put Everton 2-0 ahead 10 minutes into the second half.
The Merseyside outfit were down to 10 men from the 34th minute following Kevin Mirallas's dismissal for a second bookable offence, and after Lukaku missed a second-half penalty, they were punished by the visitors.
The Merseyside outfit started brightly as Adrian was forced into making a save in the seventh minute when James McCarthy's half-volley on the edge of the area flew towards the net.
Roberto Martinez's side had a penalty shout denied shortly after the save when John Stones's shirt was tugged by Reece Oxford in the box while waiting to receive the corner kick, but referee Anthony Taylor waved away the claims.
West Ham's only real attempt inside the first 15 minutes was when Payet drilled a low shot towards the bottom left corner following a surging run from midfield, but his attempt went wide.
Everton's early dominance paid off when Lukaku's low strike bounced off the far post and into the net after he spun away from Oxford in the box in the 13th minute.
The home side created a handful of chances in the first half, one of which resulted in Adrian pulling off a save to deny Ross Barkley's goal-bound shot, but Slaven Bilic's men showed threat on the counter-attack.
Their best opportunity of the half came when Aaron Cresswell whipped the ball across the face of goal, requiring a simple tap-in from Emmanuel Emenike, but the forward failed to reach it.
Everton headed into the half-time interval with the lead, but they also returned to the dressing room with 10 men after Mirallas was sent off for a second bookable offence in the 34th minute.
The Belgian was given a harsh yellow card in the early stages of the game for simulation, but replays showed that he was clipped by Cheikhou Kouyate, and the second booking was for a late challenge on Cresswell.
West Ham were also unlucky in the first half as they could have been awarded a penalty when Ramiro Funes Mori appeared to tug back Angelo Ogbonna while the defender tried to latch on to the end of Payet's cross.
After the break, Antonio tried to find the breakthrough for the Hammers when he met the end of Payet's cross from a free kick, but his header fell inches wide of the left post.
As expected, Everton dropped deep more often due to having just 10 men on the pitch, but they continued to get bodies in the final third.
The Merseyside outfit's build-up play was also slick at times, and in the 56th minute it resulted in a goal as some neat passing between Lennon and Lukaku ended with the former cutting inside and slotting home.
West Ham needed a tactical change and they were almost dealt another blow when Everton were awarded a penalty after Alex Song brought down Muhamed Besic in the box, but Lukaku's tame shot towards the bottom left corner was stopped by Adrian.
A few minutes later, Lukaku had the opportunity to redeem himself after getting on the end of Barkley's through-ball and darting into the box, but his attempt to slide the ball into the net was blocked by Adrian.
Everton looked as though they were going to see the game out, but inside three minutes, Antonio and Sakho scored with headers from inside the box and Payet wrapped up the game in the 90th minute by slotting the ball under the keeper after latching on to the end of a knock-down.
West Ham move up to fifth in the table, one point behind fourth-placed Manchester City, while Everton remain 11th in the standings.
