The 28-year-old, who joined Marseille in 2015 from Metz, has signed a four-year deal at the Allianz Arena for a reported fee of €10million.
Bayern had earlier announced the loan arrival of Douglas Costa from Juventus and the free transfer signing of Eric Maxim Choupo-Moting, in addition to the arrival of Marc Roca from Espanyol for a reported €15m on Sunday.
"I'm very happy to be joining a great club like Bayern," Sarr told their official website. "It's a dream come true and I will do everything I can to help continue the successful history of this club."
Sporting director Hasan Salihamidzic added: "Bouna Sarr is a dynamic right-back who fits in very well with the way we approach our game. He'll help us in an important position.
"At 28, he brings stability and experience to the table and we also believe that he'll develop further at Bayern."
It is believed that Bayern midfielder Michael Cuisance will move in the opposite direction to Sarr before the close of the transfer window.
Cuisance, who saw a move to Leeds United fall through earlier in the window because of a failed medical, posted a goodbye message to Bayern on Twitter.
"Thank you @fcbayern !" he wrote.
"I learned a lot and met amazing team-mates. It's a new chapter for me but I will never forget the good memories and support from the fans."