Bayern came back quickly after falling behind in its home stadium, equalizing within two minutes and then scoring two more goals to win 3-1 in the second leg of their quarterfinal Wednesday and go through 4-2 on aggregate.
"I think it was very important we scored straight after they did," Bayern winger Arjen Robben said. "It was crucial today. It looked like a wake-up call because the first 10 minutes of the second half were a disaster the way we played. We were so slow, we were not there and you cannot do that in the Champions League - you will be punished. That's what they did, but it was a wake-up call and we scored three goals.
After Patrice Evra put United ahead with a superb volley in the 57th minute, Mario Mandzukic got the quick equalizer, Thomas Mueller made it 2-1 in the 68th and Robben secured the victory in the 76th.
"I had no time to tell my players anything or to think what we could do after we conceded that amazing goal," Bayern coach Pep Guardiola said. "I am very proud of my team and of this club. ... They have a big heart and reacted very well. As title holders, we have a lot of pressure."
The loss means United looks set to miss the Champions League next season for the first time since 1995-96. David Moyes' team is only seventh in the Premier League, with only the faintest of hopes of finishing in the top four. Its only realistic chance of qualifying for Europe's top competition was to win it this season, but now the team's last chance for a trophy is also gone.
Bayern dominated the first half but failed to make much out of its possession advantage, and Evra gave United a surprising lead. Antonio Valencia's swinging cross sailed over everyone in the area but fell to Evra, who unleashed an unstoppable shot into the upper corner.
But two minutes later, Bayern was back in the match. Franck Ribery crossed form the left and Mandzukic's diving header beat David de Gea.
"We concentrated so fabulously but to give up that goal at that moment was very disappointing," Moyes said. "I thought the players did a great job today. We made it difficult for them for long periods but at the end we just couldn't keep it."
In the 62nd, Wayne Rooney fluffed a shot while having only goalkeeper Manuel Neuer to beat and squandered a great opportunity. Rooney had not trained the whole week because of a toe injury and took an injection to play.
Bayern punished the miss in the 68th, when Mueller knocked in a low cross from Robben.
And in the 76th, Robben cut inside from the right and made it 3-1 with a low shot that clipped the foot of Nemanja Vidic and rolled into the net, with de Gea on the wrong foot.
"I knew it would be difficult, it is not easy to play against eight players in the box. We tried to create passes and to go to wings but they defended well against crosses," Guardiola said.
Bayern had failed to win its last four matches at home against English opposition but once it stepped up the pressure and the pace there was little doubt about the outcome. United remained winless in the Bavarian capital in five outings.
"I think it's great how we came back despite this little setback," Mueller said. "If we look at both matches, I think we deserved to come through."