Bayern said the striker was signed until 2027 after protracted talks between the two clubs.
It did not disclose the transfer fee, which media reports have put at around 100 million euros. That would be a Bundesliga record, smashing the 80 million euros paid by Bayern to sign Lucas Hernandez from Atletico Madrid in 2019.
"Bayern have signed Harry Kane from Tottenham Hotspur. The England striker signed a deal to June 30, 2027. He will play with number nine on his shirt," the Bavarians said.
"The fans can look forward to Harry Kane and the new season," said club CEO Jan-Christian Dreesen. "Harry was at the very top of our wish list. So we went a bit all-in but always within a framework and within financial reason."
Kane joined Tottenham's academy aged 11 before making his senior debut in 2011. He has spent most of his career at the north London club but had loan spells at Leyton Orient, Millwall, Norwich City and Leicester City.
Kane has scored a club record 280 goals for Spurs in over 430 appearances in all competitions and 58 times for his country in 84 games, making him England's all-time leading marksman.
He is also the second-highest Premier League scorer of all time with 213 goals - 47 behind Alan Shearer.
"I am happy to be here," Kane said in club interview. "I think Bayern are one of the biggest clubs in the world. I want to keep improving and pushing myself to be the very best."
"I felt it was the right step in my career. (Bayern have) a good winning culture year-on-year," he said.
Signing Kane, who also brings a wealth of experience and leadership qualities, means Bayern have finally landed a striker to replace Robert Lewandowski who left for Barcelona in 2022.
The Bavarians struggled for goals last season and only narrowly won the league title after Borussia Dortmund unexpectedly failed to win their last game at home.
Bayern kick off their Bundesliga season against Werder Bremen next week.
Kane's arrival also looks to improve his own chances of winning major silverware having failed to end Tottenham's 15-year trophy drought during his time at the club. Spurs failed to qualify for European football this season.