Cole and Gibson were dismissed by referee Anthony Taylor for similar high challenges in the match that Everton went on to win 2-1.
The West Ham striker was the first to see red for his challenge on Leighton Baines - he had earlier given them the lead in the 14th minute.
Republic of Ireland international Gibson was dismissed in stoppage-time after Victor Anichebe and Steven Pienaar had scored to send Everton on their way to three vital points.
David Moyes was quick to bemoan the performance of the match officials in the aftermath of an important win for his side, with an appeal against Gibson's red card broadly hinted at.
"The Blues will submit the paperwork on Christmas Eve and due to the FA protocol it does mean that Gibson will be able to participate in the Boxing Day clash with Wigan Athletic," read a statement released on the club's official website.
"Everton's case will be heard on Thursday 27 December and expect to hear the outcome of the appeal later this week."
Cole's appeal will be heard on the same day, but West Ham are not in action on Boxing Day after their derby clash with Arsenal was called off because of industrial action on the tube.
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