The 24-year-old winger, who had spent four seasons at the Etihad Stadium, has signed a five-year contract with the Bundesliga champions.
A switch to the Allianz Arena for Sane had been in the pipeline for some time, with Bayern first making their interest known during the summer of 2019.
An untimely injury scuppered those plans, with a serious knee problem picked up in a Community Shield clash with Liverpool.
Sane has, however, worked his way back to full fitness and will now be returning to his homeland to open the next chapter in his career.
He has told Bayern’s official website: “FC Bayern is a great club with big goals – and these goals suit me as well. I’m looking forward to the new challenge and can’t wait to train with the team.