The Croatia international passed a medical in Madrid on Thursday before signing a four-year deal for an undisclosed fee believed to be in the region of 22million euros (£17.5m).
"I'm very happy to be here, I know a lot of things about Atletico, I've watched a few of their games and I like the team's style," Mandzukic told reporters.
"Their coach is very good, they have great players and I believe I can do great things with this team."
Mandzukic is Atletico's most expensive signing since Radamel Falcao moved to the club from Porto in 2011 and his task will be to fill the void left by last season's top scorer Diego Costa, who has left the Primera Division champions to join Chelsea.
Atletico's sporting director Jose Luis Caminero said the club had chosen Mandzukic "because he has the characteristics we were looking for in this position".
"He is a strong striker, he is good in the air and has a potent shot," the former Atletico forward told the club's official website.
"He will allow us to pressure the opposition's defense from the start of games, because he works very hard at pressing the defense when they bring the ball out.
"He combines well with his attacking team-mates and he is used to competing at the highest level. He is a winner, a player accustomed to winning titles. He will bring a lot to our team."
Mandzukic, who played in the World Cup in Brazil with Croatia until they were knocked out in the group stage, will resume his summer holidays before joining up with the rest of the Atletico team for pre-season training later this month.
Mandzukic scored 33 league goals for Bayern in his two years with the club, moving to the Bavarian giants from Wolfsburg in 2012.
Although his goals helped the club win an unprecedented treble in 2013 of the Champions League, Bundesliga title and German Cup and a league and cup double last season, the Croatian has been deemed surplus to requirements at Bayern, with Poland striker Robert Lewandowski set to lead Pep Guardiola's side's attack next season after joining from Borussia Dortmund.
"We would like to thank Mario Mandzukic for two great years here in Munich," said the club's deputy chairman Jan-Christian Dreesen.
"We have won a total of seven titles with Mario in this time and his goals were often decisive.
"We wish him all the best both on the field and off it with his new club."
The writing appeared to be on the wall for Bayern's top goalscorer last season when he was left out of the squad for the final of the DFB-Pokal in May.
Atletico are expected to make another move for a striker later this summer to replace David Villa, who has moved to MLS side New York City after a solitary season at the Vicente Calderon.
Diego Simeone's side are believed to be interested in taking Romelu Lukaku on loan from Chelsea but the money they have spent on Mandzukic appears to be a signal that they have given up on their pursuit of Manchester City forward Alvaro Negredo, who is valued at over £25m.
Heading the other way, out of Atletico, is Brazilian midfielder Diego Ribas after Turkish club Fenerbahce announced the signing of the 29-year-old.