The Baggies were awarded a late penalty after Mike Dean adjudged Calum Chambers to have handled the ball in the area, and Jay Rodriguez subsequently converted to earn the home side a point at the Hawthorns.
Wenger angrily confronted Mike Dean and his match officials on the pitch, along with his goalkeeper, Petr Cech, to question the decision, while it is reported that he also entered the referees' room after the game.
That action is forbidden unless done so with express permission, and the FA has now punished the Frenchman for his actions.
The FA have released the following statement: "Arsenal manager Arsene Wenger has been charged following the West Bromwich Albion game on Sunday [31/12/17].
"It is alleged his language and/or behaviour in the match officials' changing room after the game was abusive and/or improper and/or questioned the integrity of the match referee.
"He has until 6pm on Friday [05/01/18] to respond."
Wenger could be handed a touchline ban, after he was suspended from the touchline for four matches last season after attempting to push Anthony Taylor during Arsenal's 2-1 win over Burnley in January 2017.
His previous behaviour may, subsequently, count against him this time around.