The 24-year-old was instrumental in Germany's World Cup-winning campaign that culminated in a 1-0 final victory over Argentina on Sunday and moves to the Bernabeu for a fee believed to be in the region of 25 million euros (19.7 million pounds).
Kroos is Real's first signing ahead of the new season and he could well take over set-piece duties from Xabi Alonso. His arrival could also lead to the departure of Germany team mate Sami Khedira, who has been linked with a move to England.
"We are grateful to Toni Kroos for his time in Munich," Bayern chairman Karl-Heinz Rummenigge told the club website.
"Together we had great achievements here and we wish he and his family all the best in Madrid with Real."
Kroos joined the Bayern youth setup in 2006 and was a key member of the back-to-back domestic double winning sides of the past two seasons following a short loan spell at Bayer Leverkusen.
(Editing by John O'Brien)
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