Lewandowski completed his hat trick in the last minute, just after Leon Goretzka put Bayern ahead for the first time. Erling Haaland had fired the visitors into an early two-goal lead.
"We had a bit of luck in the end but it was deserved, because we were the dominant team," Bayern coach Hansi Flick said.
The win kept Bayern two points clear of Leipzig, which increased the pressure on the defending champions earlier with a 3-0 win at Freiburg. It was Leipzig's sixth straight league victory. Ten rounds remain after this weekend.
It was the sixth consecutive league game in which Dortmund conceded at least four goals in Munich, a bitter blow for Edin Terzić's team after frustrating Bayern for the first 88 minutes.
Dortmund complained that Leroy Sané fouled Emre Can in moving him off the ball in the buildup to Goretzka's goal, but referee Marco Fritz allowed play to continue.
"If it was a Bayern player it would have been a foul. That's just the way it is," Dortmund captain Marco Reus claimed.
The visitors made a furious start. Haaland scored in the second minute with the help of a deflection from Bayern defender Jerome Boateng, seven minutes before Thorgan Hazard set Haaland up for a tap in.
But Lewandowski pulled one back in the 26th and equalized from the penalty spot before the interval after Mahmoud Dahoud fouled Kingsley Coman in the area.
Haaland hurt his right ankle after a foul from Boateng and had to go off in the 60th. Boateng was carried off 10 minutes later after injuring himself making a clearance.
Bayern pushed harder for the win with Dortmund relying on long balls forward for reserve team striker Steffen Tigges. The visitors were without Jadon Sancho, Gio Reyna and Raphaël Guerreiro due to injury.
Goretzka reacted quickly to Thomas Meunier's attempted clearance to score in off the left post, and Lewandowski claimed his 31st goal of the season while Dortmund was still reeling.
"We started good but then ... we didn't play football anymore," Can said. "It was difficult of course. We have to do more. That wasn't good enough today."
HERTHA CELEBRATES
A late penalty from Dodi Lukebakio earned Hertha Berlin a 2-1 win over Augsburg to end its nine-game run without a win.
Hertha coach Pál Dárdai – who went into the match still looking for his first win in his second stint in charge – had never lost in seven previous games against Augsburg, but his team got off to a bad start with Laszlo Benes scoring with the visitors' first attack in the second minute.
Florian Niederlechner hit the post with the visitors' second attack in the 38th.
Hertha found gaps hard to come by against Augsburg's well-drilled defense. Rafael Gikiewicz also made some crucial saves to deny Hertha.
Krzysztof Piatek finally equalized in the 62nd with a fine header to Vladimir Darida's cross, and Hertha was fortunate to be awarded a penalty when Lucas Tousart went down – after some delay – under Mads Pedersen's challenge. A VAR check confirmed the decision and Lukebakio scored the winner from the spot in the 89th.
It lifted Hertha to 14th, three points from the relegation zone.
GLADBACH 'ROSE-DIVE'
Patrik Schick's 77th-minute strike on a counterattack was enough for Bayer Leverkusen to win 1-0 at Borussia Mönchengladbach, dealing the home team its fifth defeat in five games since it was announced that coach Marco Rose is leaving for Dortmund next season.
Leverkusen was also desperate for points. Peter Bosz' team had only one win from its last nine games, which included defeats that saw his team knocked out of the German Cup and Europa League.
Third-place Wolfsburg slumped to a 2-1 loss at Hoffenheim, and Filip Kostic rescued a 1-1 draw for fourth-place Eintracht Frankfurt against visiting Stuttgart.
