Dutchman Arjen Robben scored a sublime volley late in the second half at Old Trafford to cut United's lead to 3-2 on Wednesday, sending Bayern through on away goals.
This was not the first time Bayern, who will next face Olympique Lyon, had defied the odds to advance this season.
"You can only hand out a huge compliment to the team," Bayern general director Karl-Heinz Rummenigge told reporters early on Thursday after his team made the last four for the first time since winning the trophy in 2001.
"You can only take your hat off and bow to the team's determination, energy and passion."
Bayern's progress to the last four came as a surprise, given the Bavarians' shaky start to the competition where they were all but eliminated on several earlier occasions.
Bayern, who scored in stoppage time to win the first leg against United 2-1 after going a goal down, battled back from 3-0 this time, with Ivica Olic scoring just before halftime and Robben adding the crucial second away goal in the 74th minute.
"We have to remember that we were almost out of the competition a few times this year," said Rummenigge.
"Against (Juventus) in Turin or in Florence (against Fiorentina) and again here in Manchester. But this team has an unbelievable will and amazing passion. Obviously with such attributes you can go very far in this competition."
Bayern needed to win their last group match against Juventus to advance at the expense of the Italians. They crushed them 4-1 in Turin to progress.
They then edged out Fiorentina in the round of 16 thanks to a controversial last-minute goal in Munich and an away goal scored again by Robben in Italy.
"After we went 3-0 down against Manchester I also thought this would be tough here. But even before the match we said that a lot is possible in this season," Robben said.
Bayern now face Lyon, who came out on top in their all-French tie against Girondins Bordeaux, with the first leg on April 21 in Munich and the return leg on April 27 in France.
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