Bruma, 22, has previously been linked with a move to Manchester United, but Bundesliga runners-up Leipzig agreed a £12m deal with Galatasaray for the transfer of the highly-rated winger.
"Bruma fits perfectly with our football and the physical facilities to our game," Leipzig sporting director Ralf Rangnick told the club's website.
"He is fast, technically strong and has an extreme pull to the goal. We are glad that we could sign a player with such talent for us. Bruma will expand our options in the offensive game and give us even more flexibility in the number 10 position."
Bruma, who started his professional career with Sporting Lisbon, scored 11 times and provided six assists in 30 Super Lig appearances for Galatasaray last season.