Kompany, 38, joins Bayern on a three-year deal despite seeing his Burnley team relegated from the Premier League this season.
"I'm looking forward to the challenge of FC Bayern. It's a great honour to be able to work for this club - FC Bayern is an institution in international football," said Kompany in a club statement.
"I'm now looking forward to the basics: working with the players, building a team. Once the basis is right, success will follow."
Kompany's hiring brings to an end Bayern's exhaustive search for a replacement for Thomas Tuchel after the club's first trophyless season in 12 years.
Bayern announced that Tuchel would depart at the end of the season in February after falling behind eventual champions Bayer Leverkusen in the title race. Leverkusen coach Xabi Alonso, Germany boss Julian Nagelsmann and Austria manager Ralf Rangnick all reportedly turned down Bayern's interest.
After taking over at Burnley in July 2022, Kompany led the Clarets to promotion from the second-tier Championship in his first season in charge. They struggled with the step up to the Premier League, winning just five games and registering 24 points - the club's lowest top-flight tally in their history.
However, Kompany retained the backing of his old coach at Manchester City Pep Guardiola, who advised his former club to go for the Belgian.
"Vincent Kompany is the type of coach who fits in very well with FC Bayern's playing philosophy and identity," said sporting director Christoph Freund. "He's a young, very ambitious coach with a lot of international experience.
"He his finger on the pulse when it comes to the players and knows exactly what needs to happen on the pitch. He can and will give our team an enormous amount."
Kompany has Bundesliga experience, having played at Hamburg for two seasons early in his playing career and his ability to speak German was seen as a major benefit for the Bayern board.
Burnley will reportedly receive £10 million ($13 million) in compensation for allowing their manager to leave.
"Whilst we were initially confident in retaining our manager, the changing dynamics of the situation made this impossible," Burnley said in a statement.
"We understand the allure and prestige of a club like Bayern Munich and respect Vincent’s ambition to explore new opportunities."