The 23-year-old trod on Ibrahimovic's head during the first half of Bournemouth's 1-1 draw at United last Saturday.
Bournemouth said they were "extremely disappointed" with the decision, which they described as "extraordinary".
Mings, 23, trod on Ibrahimovic's head during the first half of Bournemouth's 1-1 draw at United in the Premier League last Saturday.
Ibrahimovic retaliated by elbowing Mings in the head, for which he accepted a three-game ban on Tuesday.
"AFC Bournemouth's Tyrone Mings will serve a five-match suspension, following an Independent Regulatory Commission hearing today," the FA said in a Twitter post.
"Mings was charged with violent conduct in relation to an incident in or around the 44th minute of the game against Manchester United."
A Bournemouth statement said: "We will study the detailed reasons of the commission once they become available, but find it extraordinary that the charges can be described as 'proven' when there is absolutely no evidence to prove the incident was intentional.
"It is our strongly held belief –- backed up by our relationship with the player and knowledge of his background and character –- that it was an accidental collision.
"Tyrone twice apologised to Ibrahimovic during the match for the accidental collision and also reiterated that there was no intent straight after the final whistle in a series of television interviews."
Match referee Kevin Friend did not spot either incident involving Mings and Ibrahimovic.
As a result, they were referred to a panel of three former elite referees, who unanimously concluded that both offences merited red cards.
Mings denied deliberately stamping on Ibrahimovic, saying: "That's not in my game. Hard and fair is how I like to tackle, but off-the-ball stuff like that isn't part of my game."
He will miss Bournemouth's league games against West Ham United, Swansea City, Southampton, Liverpool and Chelsea and will not be available until the trip to Tottenham Hotspur on April 15.