The 29-year-old striker was handed the punishment after being found guilty of breaking Scottish Football Association regulations which forbid any gambling on matches around the world.
Moffat was found guilty of placing seven bets on games involving his club but Ayr chairman Lachlan Cameron branded the punishment "grossly unfair" after comparing it to the ban given to Rangers midfielder Ian Black.
Former Inverness and Hearts man Black received an immediate three-match suspension for several betting offences including placing three bets on his then team not to win, something Moffat was not accused of - he had bet on Ayr to win.
A statement on the Scottish League One cluib's website read: "After taking the opportunity to sleep on it the club has decided to appeal the decision and sanctions handed down to Michael Moffat at yesterday's tribunal.
"The club does not take gambling lightly and are committed to working with the SFA to help eradicate gambling from the game. However, we feel that the decision yesterday was grossly unfair based on the circumstances and that there are inconsistencies in the decision-making process that led to the panel's decision.
"We will be appealing under rule 14.8.4 that states 'the sanction(s) imposed by the tribunal was excessive or inappropriate'. We feel that we have a legitimate case here and look forward to the chance to reverse the decision and have it replaced by a much more appropriate sanction."