Patrice Evra's sensational strike put the visitors ahead after the pause, but Mario Mandžukić extinguished the Frenchman's opener less than two minutes later before Thomas Müller pounced to put Bayern ahead in the tie. Arjen Robben eventually ended United's valiant effort.
United will not play Champions League football for at least 17 months
Here are five talking points...
WENGER RIGHT ABOUT BAYERN'S VULNERABILITY
Logically, the acquisitions of Mario Götze and Thiago Alcântara to a treble-winning squad that remained largely intact should make said squad better. Pep Guardiola is rightly regarded as one of the best coaches in the world but, like Sir Alex Ferguson, he is perhaps his club's biggest liability as well as its strongest asset. Bayern were ponderous, predictable and - startlingly - weak in numbers against the worst United squad in a decade, missing the ineligible Juan Mata and the injured Robin van Persie and Rafael da Silva. Four of Bayern's substitutes were unknowns, and although the absent Thiago, Bastian Schweinsteiger and Javier Martínez contributed to a shift in formation, little had changed from the first hour at Old Trafford last week. Again, it took a United goal to instigate a proper performance.
It was easy to dismiss Arsène Wenger's suggestion last month that Bayern were "vulnerable" as an embittered barb after Arsenal were eliminated, yet following a tighter tie it is difficult to disagree. United will rue their heavy touches in the final third and Wayne Rooney's profligate finishing; Moyes' side gave an inkling as to how this Bayern side can be bettered in a European tie.
The man for the big occasion: Thomas Müller scored another big goal
Bayern are not as entertaining as Jupp Heynckes' merciless marvels of a year ago, even though Guardiola could emulate his predecessor's feat. The restrictions imposed are solely the coach's doing.
EVRA THE LIABILITY
How appropriate Patrice Evra, whose blockbuster strike past Manuel Neuer put United ahead, was culpable for Mario Mandžukić's leveller 22 seconds after the re-start. The Frenchman was effectively still celebrating his cracking half-volley, but his lack of appetite for defending has been evident all season.
There was little effort to close down Arjen Robben for Thomas Müller's winner and he was easily dismissed by the lightweight and calculable Dutchman for the clincher. Ferguson's failure to leave Moyes with a quality alternative at left-back is almost as unfathomable as his midfield blind spot.
Evra struggled to cope with Robben, who made the difference
CARRICK FAILS TO TAKE RESPONSIBILITY AGAIN
At his United zenith, Michael Carrick had one Italian newspaper gushing "bellissimo Carrick" following his masterclass at San Siro five years ago. Those 90 minutes against Internazionale now represent a glimpse of the player he should have emerged into at United.
Ferguson's aversion to signing midfielders in his last six years as manager restricted Carrick, who papered over that area's cracks excellently over the last two seasons, but he has experienced arguably his worst season at Old Trafford and was wasteful in possession again at the Allianz Arena. In a team of pretenders aiming to register an upset, his maturity and knowledge of playing at Bayern four years ago ought to have provided a more composed display, as the artisan Darren Fletcher was tasked with the donkey work. Unfortunately for United, it was a tentatively recognisable effort from the 32-year-old.
ROBBEN PREDICTABLY BRILLIANT
He receives the ball, he cuts inside and he shoots. He will rarely pass and he will sometimes score. No, it is not Andros Townsend, but the player he aspires to be. Arjen Robben's avarice on the ball visibly infuriated some of his teammates in the first 45 minutes, although, as is his wont, he decided this tie. His cross for Thomas Müller's winner was deflected - as was his goal - yet there is much to admire about Robben's resilience, which he exhibited in last year's Champions League final.
Moyes' first season will end trophyless
MOYES REACTIVE, RATHER THAN PROACTIVE
David Moyes shares some unwanted traits with the man he succeeded, and on Wednesday evening it extended to his substitutions. Nicknamed "Dithering" by Evertonians, he was certainly that after Mandžukić's equaliser, which galvanised a hitherto moribund Bayern side, whose dominance inevitably increased. Danny Welbeck was reliably unreliable, Darren Fletcher nor Michael Carrick could retain possession and yet Moyes waited. By the time he reacted, it was too late and two minutes after Javier Hernández was belatedly introduced Robben scored Bayern's third.