Cole and Gibson win red card appeals
Cole was dismissed by referee Anthony Taylor for a high challenge on Leighton Baines and Gibson also saw red for a similar challenge on Mark Noble during Everton's 2-1 win.
Everton appealed against Gibson's three-match ban on Christmas Eve and announced on their website on Thursday that the dismissal had been overturned at a Football Association hearing.
Cole has also seen West Ham's appeal over his red card upheld by the FA, meaning he will be available for Saturday's Premier League match against Reading.
"Both players were sent from the field of play in the Premier League fixture between West Ham United and Everton at Upton Park on Saturday 22 December.
"Their three-match suspensions have been withdrawn immediately and written reasons will be provided at a later date."