Sane, 24, has agreed a five-year contract with the Bundesliga champions, who confirmed the transfer on Friday.
The Germany international will initially cost Bayern €49million ($55m), rising to €60m ($67.4m), Stats Perform News understands.
Bayern CEO Karl-Heinz Rummenigge said: "We're very happy to welcome Leroy Sane to FC Bayern. He is an outstanding player who has proven his qualities over the past few years, especially in the national team.
"Our goal is to gather the best German players at FC Bayern and the signing of Leroy emphasises this goal. I would like to congratulate Hasan Salihamidzic on successfully concluding the transfer."