The 23-year-old left-back has signed a contract with the Bavarian club running until 2031.
Eintracht Frankfurt receive a fee of €55million including bonuses for the defender, a significant financial windfall for the Bundesliga club. It represents a notable sporting loss for Frankfurt, but also an immense financial gain, having paid just €5.5million to sign Brown from 1. FC Nurnberg.
Brown developed into a full international during his time at Frankfurt, and now takes the step up to Germany's record champions.
He impressed at the World Cup this summer, registering a goal and an assist in Germany's 7-1 win over Curacao before starting further matches for his country during the tournament.
At Bayern Munich, Brown becomes the fifth most expensive signing in the club's history, behind Harry Kane, Lucas Hernandez, Luis Diaz and Matthijs de Ligt.
Bayern's recently unveiled signing Ismael Saibari also cost €55million including add-ons, matching Brown's fee. Bayern sporting director Max Eberl explained the thinking behind the move.
"We are very pleased to bring Nathaniel Brown to Bayern Munich, a player who has developed rapidly and still possesses a great deal of potential, with the future ahead of him," Eberl said.
"He made his mark at the World Cup and showed why we have been following him for some time. He will expand our options in defence and midfield."
Brown himself explained what the move means to him.
"Bayern Munich are one of the best clubs in the world," he said.
"The chance to play for this club means an incredible amount to me and fills me with pride. I also come from Bavaria, which makes it even more special.
"I have always dreamed of playing at the highest level for the biggest titles, I set myself the highest goals and want to get the maximum out of myself every day.
"In the discussions with those in charge, I quickly had a great feeling. Vincent Kompany in particular showed me how he sees me and my development, and he completely convinced me."
Brown's arrival adds further quality to Bayern's defensive options as they look to build on last season's Bundesliga title triumph under Vincent Kompany.