Wayne Rooney made a surprise comeback from the ankle damage he sustained in the first leg at the Allianz Arena last week but it was Bayern's Dutch winger Arjen Robben, not the United talisman who was substituted early in the seccond half, who decided the tie with a sumptuous volley 16 minutes from time.
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It was an unlikely triumph for the Bundesliga leaders who seemed as stunned as every one inside the stadium when England striker Rooney ran out ready for action at the start of a sensational night of European action at the Theatre of Dreams.
Seven tumultuous minutes was all it took for United to turn a 2-1 first leg deficit into a 3-2 aggregate advantage with goals from Darron Gibson and Nani - their first in the Champions League.
Bayern looked dead and buried when Nani struck again after 41 minutes but Croatian Ivica Olic, scorer of the late winner in Munich, grabbed one back two minutes later and when United's Brazilian full-back Rafael was sent off five minutes after the restart and a tired-looking Rooney was replaced the German side took charge.
Former Chelsea winger Robben struck the killer blow, drilling an unstoppable volley past Edwin van der Sar to set up a semi-final against Olympique Lyon.
United's defeat means no English clubs have reached the last four for the first time since 2003.
It also leaves United's season in danger of imploding after their costly defeat by Chelsea last Saturday means they are no longer favourites to win the Premier League.
Whereas United were stale and lacking passion against Chelsea in the absence of the prolific Rooney, they were unstoppable in the first half on Wednesday despite the England frontman clearly not firing on all cylinders.
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Bayern appeared so rattled by Rooney's presence that they forgot the basics of defending in an opening United salvo.
Rooney's first meaningful touch of the ball released Gibson and he was allowed the freedom of the Bayern half before cracking a right-foot shot inside Hans-Jorg Butt's post.
Four minutes later, with panic breaking out in the Bayern ranks, Antonio Valencia teased Holger Badstuber near the right touchline and delivered a low cross that Nani met with an instinctive back heel to lift the roof off the stadium.
When Nani curled a fantastic shot high into the net after 41 minutes it seemed a repeat of the 7-1 rout of Roma at the same stage three years ago was on the cards when United also overturned a 2-1 first-leg deficit.
