The 24-year-old winger - who is a German international - has made the switch from Manchester City to the Bavaria in a deal worth around an initial £44.7 million, according to the BBC.
Speaking after making the move, Sane said: "I want to win as many trophies as possible with Bayern, and the Champions League is the top priority."
Sane has enjoyed a successful spell with Manchester City, but has failed to win the European cup during his time in England.
However, Bayern have finally landed their long-term target and are now targeting even more success at home and abroad.
Bayern chief executive Karl-Heinz Rummenigge said: "Our goal is to gather the best German players at FC Bayern and the signing of Leroy emphasises this goal."
During his injury-hit time at City, Sane managed to score 39 goals and make 45 assists.
He was an integral member of Pep's Guardiola's team at one stage, but injury issues have seen him seemingly slip down the pecking order.
Sane will begin training with Bayern next week, though he won't be available to them during the Champions League run-in this season.