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.
"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."
"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.
