Bayern reportedly agreed a compensation fee of £10.2million with the Clarets for Kompany.
“I’m looking forward to the challenge of FC Bayern,” said Kompany.
“It’s a great honour to be able to work for this club – FC Bayern is an institution in international football.
“As a coach, you have to stand for what you are as a character: I love having the ball, being creative – but we also have to be aggressive and courageous on the pitch.
“I’m now looking forward to the basics: working with the players, building a team. Once the basis is right, success will follow.”
He is tasked with regaining the Bundesliga title after Bayer Leverkusen ended Bayern’s 11-season stranglehold on the German league.
Bayern chief executive Jan-Christian Dreesen: “All of us at the club agree that Vincent Kompany is the right coach for FC Bayern and we are very much looking forward to working with him.
“Max Eberl (board member) and Christoph Freund (sporting director) have worked tirelessly and meticulously to find a coach who will lead FC Bayern to success and continue to develop the club with new ideas.
“Vincent Kompany represents exactly the togetherness and team spirit that we need.”