“FC Bayern is a very big club and has big goals – these goals suit me as well,” said the 24-year-old, who added wants to win the Champions League with the Bundesliga giants.
“I’m looking forward to the new challenge and can’t wait to train with the team.”
Sane is looking forward to linking up again with Bayern head coach Hansi Flick, who has just steered the club to an eighth straight league title in his first season in charge.
“I know Hansi Flick from the U21 national team, we had a very good relationship there,” added Sane.
“I want to win as many titles as possible with FC Bayern, and the Champions League is at the top.”
The Bundesliga champions did not give the transfer fee, but Sky Sports and the BBC have reported that Bayern and City agreed a fee of £54.8 million (59.97 million euros, $67.36 million).
German daily Bild claim the fee is around “at least 50 million euros”.
“We are happy that we can welcome Leroy Sane to Bayern,” said club chairman Karl-Heinz Rummenigge.
“He is an excellent player who has proven his qualities over the past years, especially in the national team.
“Our goal is to gather the best German players at FC Bayern and Leroy’s commitment confirms that.”
The 24-year-old will start preparing for next season in Munich next week, the club has said, meaning he will sit out Saturday’s German Cup final against Bayer Leverkusen in Berlin.
Last year, Sane had reportedly been on the verge of a three-digit million euros transfer to Bayern before suffering a serious knee injury in August which sidelined him until February.
After rebuilding his fitness, the Germany international made his first Premier League appearance of the season as a substitute against Burnley last week.
His transfer became an open secret after City manager Pep Guardiola revealed earlier this week that Sane was likely to leave after rejecting a new contract offer with a year left on his contract in Manchester.
“We wish Leroy all the best in this new chapter of his career,” said Guardiola in a statement on Friday.
“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.”
Sane, who won the Premier League title twice with City, has scored 39 goals in 135 matches in all competitions for the Citizens since joining from Bundesliga club Schalke in 2016.
The fleet-footed winger will bolster Bayern’s attack out wide.
“With Serge Gnabry, Kingsley Coman and Leroy, we are now very well occupied at the highest level on the wing, positions that are important for our game,” said Bayern sports director Hasan Salihamidzic.
Sane is Bayern’s third new signing for next season alongside French teenage defender Tanguy Nianzou Kouassi from Paris Saint-Germain and goalkeeper Alexander Nuebel from Schalke, both of which were free transfers.