Since the Champions League final defeat to Real Madrid on May 29, Mane has been heavily linked with an exit after a six-year stay at Anfield.
Bayern soon emerged as the most likely destination for the Senegal international, despite frequent reports claiming that they were well below Liverpool's valuation.
However, the Merseyside giants confirmed the signing of Benfica forward Darwin Nunez last week, effectively ensuring that a departure for Mane was only a matter of time.
The Bundesliga champions have now followed suit, revealing that Mane has penned a three-year contract in a deal which could cost £35m.
Speaking to the club's official website, Mane said: "I'm really happy to finally be at FC Bayern in Munich. We spoke a lot and I felt big interest from this great club right from the beginning, so for me there were no doubts.
"It's the right time for this challenge. I want to achieve a lot with this club, in Europe too. During my time in Salzburg I watched a lot of Bayern games - I really like this club!"
Herbert Hainer, the Bayern president, added: "Sadio Mane is a global star, who underlines the appeal of FC Bayern and will increase the attractiveness of the Bundesliga as a whole.
"Fans come to the stadiums to see such players. It's great that our board led by Oliver Kahn and Hasan Salihamidzic have managed to sign a player like Sadio Mane for FC Bayern."
Since signing for Liverpool in 2016, Mane, under the guidance of Jurgen Klopp, has progressed into one of the best forward players in world football.
The attacker contributed 120 goals and 48 assists from his 269 appearances in all competitions, a return of 22 goals in the 2018-19 Premier League campaign representing his best effort statistically.
Mane's regular availability has perhaps gone unnoticed, the player failing to feature in just 12 top-flight fixtures during the last four seasons.
During his time on Merseyside, Mane has won all of the major trophies, the standout achievements being the 2018-19 Champions League and the Premier League title triumph which followed in 2019-20. body check tags ::