Already-qualified Bayern, who still secured top spot in Group E, were two goals ahead at the break thanks to a double strike from Mario Gomez but the Romans marched out for the second half intent on bombarding the opposing goal.
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Three incisive passes pierced Bayern's defence and led to goals from Marco Borriello, Daniele De Rossi and a Francesco Totti penalty as the Stadio Olimpico faithful roared with delight at an eighth game unbeaten in all competitions.
Coach Claudio Ranieri's empire was crumbling at the start of the season but a remarkable turnaround in form means Roma will guarantee themselves a place in February's first knockout round with a draw at CFR Cluj in the final group game on December 8.
"The lads were magnificent, they ran, they fought and they were helped by the fans who were marvellous," Ranieri told Sky.
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The coach introduced Totti with 15 minutes left and the substitute played in Borriello with a beautiful ball that led to the striker being debatably felled by inexperienced goalkeeper Thomas Kraft, who was unlucky not to save the winning penalty.
"Totti had played two games so I let him rest a bit but I knew I could count on him, he is a stupendous athlete," Ranieri said of the 34-year-old captain.
BOOED OFF
In-form Gomez, who blasted a hat-trick in the previous group win over Cluj to secure progression, had lit up the Eternal City in the first period and looked to have put last term's runners-up on track for a fifth win from five European games.
He finished off a hopeful clip across the box from Franck Ribery, starting for the first time since injury in September, after 33 minutes and six minutes later he had the ball in the net again after poor defending involving Roma's Jeremy Menez.
The hosts were booed off at the break but the French playmaker redeemed himself when his run and cross was turned in by Borriello, albeit from the ground after the ex-AC Milan man fell over while controlling the ball.
Another superb pass from Mirko Vucinic set up John Arne Riise to cross for Daniele De Rossi to score from close range after 81 minutes before Totti's spot kick secured the win.
Bayern coach Louis van Gaal, recently criticised by Ribery and club boss Uli Hoeness for being arrogant with the team struggling eighth in the Bundesliga, was fuming.
"We gave the match away like a present. It is unbelievable, we were so dominant. Then our concentration was gone. This defeat is not good for our self esteem," he told German TV.
"The problem lies in our attack, not defence. It begins with losing the ball but my players don't seem to understand."
