The 27-year-old has signed a five-year-deal with Die Werkself and will make the move to the BayArena at the end of the 2021-22 season.
Azmoun has been linked with a number of clubs in recent weeks, including Lyon and English teams Burnley, West Ham and Newcastle, but the Bundesliga side has won the race for the Zenit St. Petersburg forward.
The Iran international has scored 62 goals in 104 games for the Russian club.