The Frenchman was charged with misconduct for his behaviour towards the referee during the Gunners' 1-1 draw with West Bromwich Albion.
Wenger has accepted the charges and will now miss the Gunners' next three encounters.
A statement from the FA read,"Arsène Wenger will serve a touchline ban for Arsenal’s next three matches after he was charged with misconduct.
"It follows an Independent Regulatory Commission hearing today at which the Arsenal manager admitted that his language and behaviour in the match officials’ changing room after the West Bromwich Albion game on Sunday [31/12/17] was abusive, improper and questioned the integrity of the match referee. He was also fined £40,000."