Haaland drove Dortmund ahead from 20 yards after just 74 seconds and then turned home Thorgan Hazard’s cut-back.
But Lewandowski steered in Leroy Sane’s cross and levelled from the spot after Mahmoud Dahoud fouled Kingsley Coman.
Leon Goretzka drilled Bayern ahead late on before Lewandowski’s third confirmed a return to the top of the Bundesliga.
Victory moved Hansi Flick’s reigning champions two points clear of second-placed RB Leipzig, who had briefly climbed above them by winning at Freiburg earlier on Saturday.
Dortmund, meanwhile, drop a place to sixth, four points off the Champions League places, their woes compounded by a knock to Haaland that forced the Norway striker off in the second half.
He joins England winger Jadon Sancho on the treatment table, the England winger absent with a thigh injury, and Dortmund will be desperate for both to be fit by Tuesday as they bid to protect a 3-2 first-leg advantage against Sevilla in the Champions League last 16.
BBC Sports
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