The 28 year-old Austria international emerged as a deadline day target for Erik ten Hag's side after an injury to Christian Eriksen that will leave the Denmark international out of action until April.
Ten Hag told reporters that United did not require a late addition to fill the void left by Eriksen.
However, the availability of Sabitzer changed that approach and he has now completed a move to Old Trafford, bolstering the squad's midfield options for the remainder of the season as they battle on four fronts.
United confirmed Sabitzer's arrival on their official website a little over an hour after Tuesday's transfer deadline, having submitted the relevant paper work in time.
"Sometimes in life you have to make quick and important decisions," Sabizter said. "From the moment I heard about this opportunity I knew it was right for me. I am a competitive player; I want to win and help the club achieve its aims this season.
"I feel that I am at my peak as a player, and that I can contribute a lot of experience and energy to the squad. I am excited to start with my new team-mates and manager and to show my qualities to Manchester United fans."
United director John Murtough said: "Marcel is a player that we have watched for a long time. The opportunity arose quickly, and we knew that he was someone with the ability and character to make an impact.
"He adds further quality to our squad and experience to the dressing room, and all of us are pleased to be welcoming him to Manchester United."
Sabitzer joined Bayern from Bundesliga rival RB Leipzig ahead of the 2021-2022 season but has found regular opportunities difficult to come by.
While he has made 40 appearances for the Bundesliga champion, just 15 of those have been as a starter.
Sabitzer has not been signed in time for the second leg of United's EFL Cup semi-final against Nottingham Forest on Wednesday, but he could make his debut in the Premier League clash against Crystal Palace on Saturday.
