Gnabry said: "It's a great honour to become part of FC Bayern".
"I have an exciting challenge ahead and I am really looking forward to it".
"We are very happy to have secured the services of another young Germany worldwide in Serge", he stated. "Serge has developed really well in Bremen".
Gnabry previously played for Arsenal but could not convince manager Arsene Wenger of his pedigree, making just 18 appearances for the Gunners and scoring once, and he was subsequently sent out on loan during the 2015-16 season, playing three matches for West Brom.
German media reported at the time that Bayern Munich were involved in the transfer and had the option to purchase the player after year, suggestions which were strongly denied by Bremen sporting director Frank Baumann.