Lewandowski, 25, will join the German champions on a free transfer when his contract at Signal Iduna Park expires at the end of the season.
The 25-year-old has agreed terms on a five-year deal at the Allianz Arena, leaving Dortmund in need of a replacement for the highly-rated No. 9.
“We’re very pleased about completing this transfer,” said FC Bayern Munich AG chairman Karl-Heinz Rummenigge.
“Robert Lewandowski is one of the best strikers in the world. He will strengthen the Bayern squad and give us another boost. We’re delighted both parties have today signed a five-year contract until 2019.”
Dortmund have been linked with a number of strikers in recent times, including the likes of Aston Villa ace Christian Benteke, Atletico Madrid sensation Diego Costa and Manchester City’s third-choice striker Edin Dzeko.
Dzeko, who joined City from VfL Wolfsburg in a deal worth €32 million in January 2011, played in Germany between 2007 and 2011, while Diego Costa is 25/1 to leave the Estadio Vicente Calderón for Signal Iduna Park.
Giaccherini plays down Juve rumours
Sunderland winger Emanuele Giaccherini has moved to play down reports linking him with a quickfire return to Juventus, Sky Sports reports.
The Italy international, 28, joined Sunderland from Juve on a four-year deal in the summer and has gone on to make 20 appearances in all competitions for the Black Cats, tallying three goals and one assist.
The move from Turin to Wearside came as shock to many, and rightfully so.
Leaving the two-time Italian Serie A champions for Sunderland is a step, or five, down the football ladder, especially when considering Giaccherini has established himself as a regular in the Azzurri squad under Cesare Prandelli.
“I feel good in England, even if things could go better,” the 28-year-old told Radio DeeJay.
“We will see in May what will happen, but for the moment the most important thing is getting safety.
“It’s normal to be missing the city of Turin and Italy, especially when you leave a club where you won two scudettos, where you felt good and that allowed you to get into the national team.
“I’d always be grateful to coach Antonio Conte. I spoke with him twice before moving to Sunderland.
“He did everything to keep me, but then the club and the player had to make certain evaluations. My close future remains in England.”
Sunderland will host League One side Carlisle United in Round 3 of the FA Cup on Sunday, 5 January 2014, with Giaccherini not expected to start.
The absence of Giaccherini could mean a start for Greece international winger Charis Mavrias, who is priced at 10/1 to open the scoring at the Stadium of Light.