The 21-year-old, who joined Die Werderaner from the Gunners last summer, following a loan spell at West Brom, scored 11 times this season.
He was also a part of the German Olympic team, which won silver medals in Brazil in 2016, and signs for the Bundesliga champions on a three-year deal.
Karl-Heinz Rummenigge, the Chairman of the FC Bayern Munich Board of Management, said Gnabry’s development at Bremen was key to them signing the player.
“We are delighted to see another young German national player in FC Bayern in Serge Gnabry,” said Rummenigge.
“Serge has developed very well in Bremen. “
Gnabry added: “It is a great honour to be part of FC Bayern. Before me lies an exciting time, which I am very much looking forward to.”
Born in Stuttgart, Gnabry represented a number of teams at youth level before joining Arsenal’s academy in 2011 as a 16-year-old.
He made his first team debut in 2012 and was making a name for himself in north London before a knee injury in 2014 hampered his progress.
In 2015 he had an unsuccessful loan spell at West Brom before moving home to Germany. He announced earlier this month his intention to leave Werder Bremen after just one season.