But referee Anthony Taylor was the pantomime villain in this match before Christmas as two needless red cards spoilt the game.
Taylor all but handed the Toffees three points after controversially deciding Carlton Cole had left his foot in on Leighton Baines midway through the second half.
The official evened up the red-card count in stoppage-time, finding Darron Gibson guilty of a similar offence on Mark Noble.
Earlier, Sylvain Distin missed a sitter to put the Toffees ahead from a free-kick from Baines, but the unmarked defender nodded it into the ground and over the crossbar.
Baines produced another special three minutes later for Leon Osman's disallowed goal, his corner headed home by one of the smallest players on the pitch but ruled out seemingly for Anichebe's block on Jussi Jaaskelainen.
The Everton players were furious and their ire only increased when Cole fired West Ham in front soon afterwards, the striker cutting inside the recalled Johnny Heitinga far too easily before cracking in a low drive from the edge of the box on 14 minutes.
At the other end, Everton continued to probe, prompting West Ham boss Sam Allardyce to replace Gary O'Neil with Modibo Maiga just before the hour mark, a move that almost paid off immediately when the Malian scuffed Cole's cross wide.
And that was when it all went horribly wrong for the hosts. Maiga got himself booked for pushing Pienaar over in the 64th minute and, from the resultant free-kick, the ball was worked left to the South African, whose cross was glanced home brilliantly by Anichebe.
Three minutes later, Taylor made the call that swung the game. Cole hung his leg out in a bid to control a dropping ball and definitely made contact with Baines but it looked anything but a reckless lunge. To Cole's horror, Taylor thought otherwise, brandishing a straight red card.
The momentum was all with Everton and they completed their comeback six minutes later, Osman's cross inadvertently played against Pienaar by Kevin Nolan before trickling into the net.
There were chances at either end before Gibson then saw red in stoppage time for an almost identical challenge to Cole's on Noble, although this one was slightly less debatable.