Taylor effectively ended the hosts' hopes 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, having earlier angered Everton by chalking off Leon Osman's early header.
Cole gave West Ham a half-time lead and Victor Anichebe levelled before Cole saw red, with Steven Pienaar making the man advantage count six minutes later.
Baines looked to have set up the opening goal when his corner was headed in by Osman but it was ruled out seemingly for Anichebe's block on Jussi Jaaskelainen.
Instead, the furious visitors fell behind when Cole netted two minutes later, the striker cutting inside Johnny Heitinga far too easily before cracking in a low drive from the edge of the box.
Nikica Jelavic struggled to trouble the Hammers backline but Everton upped the tempo after the break and made the breakthrough. Then from a free-kick in the 64th minute, the Toffees worked the ball left to Pienaar, whose cross was glanced home by Anichebe.
Three minutes later, Taylor made a 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. Taylor, though, brandished a straight red card.
Everton completed their comeback six minutes later, Osman's cross inadvertently played against Pienaar by Kevin Nolan before trickling into the net.
The numbers were evened up in stoppage time when Gibson then saw red for an almost identical challenge to Cole's on Mark Noble, although this one was slightly less debatable.