The Irishman was dismissed for a high challenge on Hammers' midfielder Mark Noble, while Carlton Cole had been sent off for the home side for a similar tackle on Leighton Baines earlier in the game.
David Moyes insisted after the game that neither player should have been shown a red card, while a Toffees statement confirmed Gibson will be available for the club's clash against Wigan on Boxing Day, due to the appeal process taking place after the game.
"I thought neither were sending offs but after the first one, he feels he has to give it," Moyes said after the game. "It was just two players honestly going for the ball. You can see by Leighton Baines' reaction, he just dusts himself down and gets on with it.
"I will appeal [Gibson’s dismissal]. I'll have a word with Sam Allardyce beforehand to see what he thinks about it, but I'm really not sure whether you could call it frivolous."
An Everton statement confirmed that the club has put the appeal in motion.
"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.
"Everton's case will be heard on Thursday 27 December and expect to hear the outcome of the appeal later this week."