Gustavo has signed a contract until June 30, 2015 with the Bundesliga champions and comes from league rivals Hoffenheim, where he has already found the net twice this season.
"We are very happy that everything has finally been cleared up," said Bayern chairman Karl-Heinz Rummenigge.
"Luiz Gustavo is the player we want and it will strengthen Bayern," he added. The player is reported to have cost 15 million euros (20 million dollars).
Gustavo will now fly to Bayern's training camp in Doha to prepare the second half of the season after a poor first half by Bayern standards that sees them fifth in the table, 14 points behind leaders Borussia Dortmund.
"Luiz Gustavo, who is eligible for matches in the UEFA Champions League, will wear Bayern's number 30 shirt," the club added.
The Brazilian is likely to be Bayern's last acquisition for a while after Rummenigge said on Sunday that the club would sign no new players in the January transfer window.
Gustavo's transfer appeared to have played a hand in the departure of Hoffenheim coach Ralf Rangnick on Saturday after Gustavo was sold to Bayern Munich over his head.
"The departure of Luiz Gustavo to Bayern during the winter break is out of the question in my eyes," Rangnick had said last month.