David Moyes' side came from behind to secure an impressive victory courtesy of second half goals from both Victor Anichebe and Steven Pienaar, with two red cards handed out
The hosts took the lead in the 13th minute after Carlton Cole took a pass in his stride before spinning his marker and burying a vicious effort into the bottom corner, before Everton equalized in the second half through a delicately glanced header by Victor Anichebe.
After Carlton Cole was sent off for a high boot after his clumsy challenge felled Leighton Baines, Everton took the lead after a superb move saw Steven Pienaar get the final touch inside the box, with Darron Gibson seeing red in stoppage time for a very similar foul to Cole's.
The hosts started the brighter of the two sides but it was Everton that went closest to opening the scoring, after Leon Osman was adjudged to have used his elbow before nodding home a Baines corner in the 11th minute.
Everton was made to pay just moments later, though, when Cole put West Ham ahead after shrugging off the attentions of Heitinga before drilling an effort into the bottom left-hand corner past the despairing Howard.
David Moyes' side raced out of the traps in the second half and went close as Baines curled an exquisite effort just wide of the goal.
West Ham should have extended the lead just after the hour mark when Cole's decent low cross found substitute Modibo Maiga unmarked in the box, but his touch was heavy and he was only able to scuff an effort wide of the post.
The visitors equalized in the 64th minute through a glanced header from Anichebe after a concerted spell of pressure, with Pienaar's ball into the ball from the left proving too good as the ball nestled in the bottom far corner.
Referee Anthony Taylor then made his second questionable call of the day after the earlier disallowed goal when he gave Cole a straight red card for a high boot on Baines, when a yellow looked the correct decision.
With a man advantage, Everton took the lead for the first time in the 73rd minute through Pienaar, after a beautiful flowing move down the left saw Tomkins clear the ball onto the South African before it inadvertently rolled past Jaaskelainen.
Gibson was giving his marching orders in stoppage time by Taylor after a high challenge, very similar to the one that saw Cole dismissed, but Everton managed to hold on to clinch an important three points.