"I'm very happy to be part of FC Bayern - it's one of the biggest clubs in the world," said Diaz, who will wear the number 14 shirt at the club.
"I want to help my new team with my style of football and my personality. My goal is to win every possible title - we will work towards that every day as a team."
Liverpool accepted an improved offer of €75m from Bayern over the weekend after initially rejecting their €60m bid.
Diaz has signed an initial Bayern deal to 2029 and will wear the No14 shirt.
Bayern's sporting director Max Eberl is proud of the deal: "Luis Díaz is an internationally experienced player with tremendous quality, great skills, and great reliability. He will help our team immediately.
"We are happy that we were able to bring him to FC Bayern. Transfers like this work when everything meshes together. Our fans can look forward to an exceptional player."
Bayern's sporting director Christoph Freund added: "Luis is a goal threat, a scorer, and at the same time, he works incredibly hard for his team. He's fast, a versatile footballer, and brings an intense style of play to the pitch.
"He has also gained extensive experience at the top level with Liverpool and the Colombian national team. Luis Díaz represents a complete package that our fans will appreciate."
Bayern Munich's chief executive Jan-Christian Dreesen also expressed his enthusiasm: "With Luis Díaz, we have succeeded in signing one of the best left wingers in the Premier League.
"He brings a true player of outstanding quality from Liverpool to Bayern – he has won titles with each of his clubs.
"As a team, we were able to implement this transfer together in a targeted manner and are looking forward to a top international player who will give our team important impetus and also enrich the Bundesliga. We would like to thank Liverpool for the fair and serious negotiations at all times."