Kane travelled to Germany on Friday night for a medical and the Bundesliga giants confirmed on Saturday he had signed a deal until June 2027 and will wear the No.9 shirt.
"I'm very happy to be a part of FC Bayern now," Kane said.
"Bayern is one of the biggest clubs in the world, and I've always said that I want to compete and prove myself at the highest level during my career.
"This club is defined by its winning mentality - it feels very good to be here."
Shortly before the deal was official confirmed, Kane posted a message on his social media accounts in which he said: "I wanted to be the first to tell you Tottenham fans that I'll be leaving the club today.
"Obviously a lot of emotions going through me right now; sad to be leaving the club I've spent nearly 20 years of my life at, from 11-year-old boy to a 30-year-old man now.
"There's been so many great moments and special memories, memories that I will cherish forever.
"I felt like it was the time to leave. I didn't want to go into the season with a lot of unresolved future talk.
"I think it's important for the new manager and the players to concentrate on trying to get Tottenham back to around the top of the table and fighting for trophies so I wish Ange (Postecoglou) and all the boys all the very best of luck.
"I'll be watching from a fan point of view now and really hope the team can be successful."
Postecoglou had revealed on Friday that Kane's move was "imminent" after a breakthrough in negotiations between the clubs was reached on Wednesday night.
It left the ball in Kane's court and he decided on Thursday to leave his boyhood club for Bayern, who will pay an initial £100 millon ($A195m) with add-ons potentially taking the deal up to an overall fee of £120m ($A235m), the PA news agency understands.
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