The pair were both shown straight red cards by referee Anthony Taylor for similar high challenges in Everton's 2-1 win.
Cole saw red for a high challenge on Leighton Baines, while Gibson later got his marching orders for a near-identical foul.
However, the FA's disciplinary panel has accepted that both players were making genuine attempts to play the ball with no intent to injure an opponent.
Their red cards for violent conduct have been rescinded and neither player will serve a ban.
Gibson played on Boxing Day while the appeal was in process - West Ham's fixture at Arsenal was postponed.
Everton face Chelsea at Goodison Park on Sunday while West Ham travel to Reading.