Referee Anthony Taylor angered Moyes by ruling out an early goal from Leon Osman, but Victor Anichebe and Steven Pienaar found the net before Gibson's harsh stoppage-time red for an alleged dangerous tackle took the shine off the win.
Moyes will seek some assistance from West Ham manager Sam Allardyce as he attempts to get the decision overturned.
"I will appeal it," he said. "I'll have a word with Sam as well, because I don't think you could put the word frivolous next to that appeal.
"Because I think you would say there was genuine reason why maybe you could appeal that.
"Now, you don't want to go and appeal and find out that maybe you get an extra game on it. That's what you don't want to do."
Carlton Cole fired the Hammers into an early lead before he was sent-off in the second-half for a similar challenge and Moyes felt Taylor was looking to even up the sides with Gibson's dismissal.
"I said I thought that wasn't the way you should referee. I think if you get one, it doesn't mean that you do the other one," Moyes said.
When asked about Osman's fine individual display, he said: "If ever there was an England player today, I thought Osman was - phew - I thought he was top notch.
"Every time he got it, he turned on the ball. He dribbled past people.
"I thought Osman was exceptional and, if he plays like that, he'll have more than one cap, I'll tell you.
"The most important thing was to get the points on the board. We've been playing like that most of the season and maybe just not turned all our good play into goals.
"But, today, thankfully we got two in the end. It was tough but we got there."