Sane, 24, has agreed a five-year contract with the Bundesliga champion, which confirmed the transfer on Friday.
The Germany international will initially cost Bayern €49 million ($79.4 million)), rising to €60 million ($97.2 million).
Bayern chief executive 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."
Sane was a Bayern target last year before injury ruled him out for much of the 2019-2020 campaign, but he is delighted to have finalised the deal, quickly revealing some lofty ambitions.
"FC Bayern is a great club with big goals and these goals suit me as well," Sane said. "I'm looking forward to the new challenge and can't wait to train with the team.
"I know Hansi Flick from the [Germany] Under-21 national team, we had a very good relationship there.
"I want to win as many trophies as possible with FC Bayern, and the [UEFA] Champions League is the top priority."
Sane will not be available for Bayern in this season's UEFA Champions League, however, with the Bundesliga champion considered among the favourites ahead of August's eight-team finals.
Meanwhile, Bayern sporting director Hasan Salihamidzic thanked City for "trusting and constructive negotiations", and the Premier League club wished Sane well.
Manager Pep Guardiola, formerly of Bayern, said: "We wish Leroy all the best in this new chapter of his career.
"Leroy has been part of one of the most special periods in this club's history and he leaves with the best wishes of everyone here at Manchester City."
City director of football Txiki Begiristain added: "From arriving as a fledgling young talent four years ago, we are proud of the contribution Leroy has made since joining Manchester City.
"He leaves with many friends and fond memories of his time at the Etihad."
Sane scored 39 goals in 135 appearances for City in all competitions, helping Guardiola's side to win the Premier League and Carabao Cup in 2017-2018 followed by a domestic treble the next season.
He addressed the City fans on his Twitter page, writing: "Thanks so much for the great time and all the support over the past four years.
"It has been a really intense and successful time which I surely won't forget."
He was also thanks by team-mates.
Leroy, I'm wishing you all the best on your next chapter in Munich. You've always been more than just a teammate for me. Will miss the time on and off the pitch together. I hope you will find such a friendly neighbour in Munich like you've had in Manchester😂🏡👌🏼 See you soon Bro pic.twitter.com/9WEuhl0hZk— Ilkay Gündogan (@IlkayGuendogan) July 3, 2020
I have enjoyed so much playing with you amigo! You’re one of the most talented players I’ve seen and I am sure you will achieve everything you want in the future.— David Silva (@21LVA) July 3, 2020
All the best @LeroySane19 pic.twitter.com/dahJ0mNPGN