A fortuitous Steven Pienaar goal gave Everton a hard-fought 2-1 victory over West Ham at Upton Park in the Premier League on Saturday.
Taylor all but handed the Toffees three points and a place back in the top four after controversially deciding 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 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 marked his Everton recall with his first goal for three months three minutes before Cole saw red, with Steven Pienaar making the man advantage count six minutes later.
Today's game was a test of how Everton and their opponents coped without their midfield heavyweights, with Toffees star Marouane Fellaini starting a three-match ban and Hammers talisman Mohamed Diame still out injured.
There certainly appeared to be a lack of drive from both sides in the opening moments and it was no surprise to see the first chance arrive via a set-piece in the ninth minute.
Baines delivered the type of devilish free-kick that is certain to keep him in the thoughts of bigger clubs with the transfer window looming, the unmarked Sylvain Distin nodding it into the ground and over the crossbar.
Baines produced another special three minutes later for 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.
Having yesterday called for clarification on "grappling" in the box, the irony surely could not have been lost on Everton boss David Moyes.
His players were furious and their ire only increased when Cole fired West Ham in front two minutes later, the striker cutting inside the recalled Johnny Heitinga far too easily before cracking in a low drive from the edge of the box.
Heitinga continued to flounder and Moyes must have been ruing his decision to switch Phil Jagielka to right-back in order to accommodate the surprise return of Phil Neville from knee surgery, the Toffees captain filling in for Fellaini.
The pace dipped again, although Osman almost provided a moment of inspiration when Nikica Jelavic just strayed offside onto the midfielder's through-ball.
Jelavic got his timing all wrong again four minutes after half-time when a rare mistake from James Collins saw the ball ricochet through to the striker, who could only send it straight at Jaaskelainen.
Gibson curled a free-kick narrowly wide and 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.
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 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. 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.
Ten-man West Ham refused to concede defeat and Nolan almost made amends by volleying narrowly wide. Several substitutions followed, including Jack Collison's first appearance of the season after knee surgery.
And just as the game was drifting towards the final whistle, it exploded into life in the dying seconds. Jelavic sent a rebound wide from barely six yards and Nolan was unable to prod home after being played in at the other end.
And Gibson then saw red for an almost identical challenge to Cole's on Mark Noble, although this one was slightly less debatable.
The Associated Press was used in this report.