Bayern have confirmed their deal with Manchester City for the attacking midfielder. Sane joins for an initial fee of €45m, rising to €50m depending on bonuses.
Sane has signed a five-year contract and stated: "Bayern is a great club with great goal.
"I like these goals. I'm looking forward to this new challenge. I still know trainer Hansi Flick from the Under-21 of Germany, we have a good relationship with each other.
"I want to win as many titles as possible with Bayern. The Champions League is top priority."