The Spanish champions have long been linked with a move for the Croatia international, who grew disenchanted under Pep Guardiola and was deemed surplus to requirements following the free transfer of Robert Lewandowski, as they sought a replacement for Chelsea-bound Diego Costa.
Bayern already announced earlier this week that they had agreed verbal terms with Atletico and Mandzukic has now signed a four-year deal with his new club after passing his medical.
"I am very happy to be here. I am well aware of Atletico and I know they have good players and a good coach," Mandzukic commented on his move.
"I know a lot about Atletico, I have seen several games and I like the style of play.
"The coach is very good, they have good players, and I think I can do great things with this team."
Atletico director Jose Luis Caminero believes Mandzukic is just what the Liga champions need in their hunt for more silverware.
"The signing of Mandzukic is great news because he has all the characteristics we were looking for," Caminero said.
"He is a strong striker with a powerful shot, who is a real aerial threat, too. He is a true winner and a player who is used to winning titles. He will add a lot to Atletico."
The 28-year-old won two Bundesliga titles, the Champions League and two DFB-Pokals with Bayern since joining the Bavarians from Wolfsburg in 2012.
He scored 25 goals in all competitions last season.