The Poland international will leave Borussia Dortmund for the reigning German and European champions once his contract expires next summer, signing a five-year deal on Saturday
The 25-year-old had long been linked with a move to the Allianz Arena side and he has now signed a five-year deal with Bayern to complete the transfer.
Lewandowski is currently the joint-top goalscorer in the Bundesliga with 11 goals, and has netted 65 times in 113 league appearances for BVB.
Clubs such as Real Madrid, Manchester United and Arsenal had also shown an interest in signing the prolific attacker, but he will continue his career in the Bundesliga instead.
Lewandowski was already close to joining Bayern last summer, but the move eventually fell through as Dortmund vetoed to a move to the Bundesliga champions.
The Poland international started his professional career at Znicz Pruszkow and also wore the jersey of Lech Poznan before joining Borussia Dortmund in the summer of 2010.
He had since developed into a key player at BVB, helping the Signal Iduna side to two Bundesliga titles, a DFB Pokal and the DFL-Supercup.
Lewandowski follows in the footsteps of Mario Gotze, who left Dortmund for Bayern ahead of the 2013-14 campaign.