The Poland international has long been linked with a move to Pep Guardiola's charges, with a close-season deal failing to materialise.
But Bayern's long pursuit finally came to an end on Saturday when the Bundesliga champions announced Lewandowski would move to the Allianz Arena on July 1 upon the expiry of his Dortmund contract.
Lewandowski's decision is sure to have disappointed Dortmund supporters, who also saw star midfielder Mario Gotze join Bayern last July.
And the former Lech Poznan man says he understands his move will not be a popular one, but hopes the fans will continue to back him for the rest of his tenure.
"We still have half a year in which we have some goals that we want to achieve together," he said in a letter to supporters on the club's official website.
"Although there are some fans who not agree with my decision, I hope for your support.
"Anyway, I'll give everything for BVB."
Lewandowski has been a bright note in a poor season for Jurgen Klopp's men, who sit 12 points adrift of leaders Bayern in fourth.
He has 16 goals in 26 appearances across all competitions so far this campaign.
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