After a goalless and tense first half, United took the lead in the Allianz Arena when Patrice Evra fired home a wonderful half-volley from the edge of the box.
But United were pegged back just over 60 seconds later as they allowed Mario Mandzukic to head home from close range.
Less than 10 minutes later, Bayern were ahead for the first time in the tie when Thomas Muller converted from the edge of the six-yard box after great work from Arjen Robben on the right.
Robben, who was United's main tormentor all evening, then decided the tie on 76 minutes as he cut inside from the right flank before sending a shot past David de Gea.
The result means that for the first time since 1995, next season will see United not competing at Europe's top table. Bayern progress to the semi-finals as they maintain their hopes of becoming the first team ever to defend the Champions League crown.
David Moyes' men spent nearly all the first period in their own half and in the end, Bayern's class was too much. The fact that the German side could field such a stellar line-up when two of their most prominent players, Bastian Schweinsteiger and Javi Martinez, were suspended shows how big a gulf remains between these two sides.
It also reinforces just how much Moyes needs to spend to turn his team into world-beaters who belong in the European elite, rather than plucky battlers.
For the first 10 minutes, United more than held their own. In fact, the small pocket of away fans inside the atmospheric Allianz Arena started to dream of a 1999-style miracle as Wayne Rooney took down a long ball from Darren Fletcher and advanced towards goal.
Rooney, showing no signs that his toe injury was hindering him, brushed Dante aside and had a window to shoot, but he dallied, seemingly caught between the idea of firing at goal and squaring to the unmarked Shinji Kagawa.
From then on, it was nearly all Bayern and Robben was a constant menace. His quick feet bamboozled Evra, but the Dutchman's curling effort flew wide.
A rare break from United ended with Antonio Valencia tapping home from close range, but his effort was ruled out for offside.
For the rest of the half, Bayern camped on the edge of the United box.
Franck Ribery rippled the side-netting with a low shot, Toni Kroos curled just over and Robben slalomed into the box only to be blocked by a wall of United defenders.
A couple of breaks almost came off for United just before the break, but the fact that Bayern had 70 per cent possession showed how dominant they were.
After the break, United were more positive. Bayern's defence backed off and Kagawa powered a low shot into Manuel Neuer's hands - the first time the German keeper had been called into action.
Then the unthinkable happened. United raced away on the counter, Valencia whipped over a beautiful ball to the back post and Evra rifled home the sweetest of half-volleys.
The Frenchman was mobbed by his team-mates and the Bayern fans fell silent, but 22 seconds later it was a different story.
United backed off as Bayern went on the attack. Ribery sent over a pinpoint cross and Mandzukic sneaked ahead of Evra to head past De Gea.
Rooney had a brilliant chance to silence the home crowd soon after but somehow he fluffed his shot deep inside the Bayern box.
Then with 23 minutes left, Bayern went ahead. Evra gave Robben the half-yard he needed to fire into the box and Muller swept the ball home.
Less than 10 minutes later the tie was over. Robben's pace proved to be too much for Evra again.
The Dutchman skipped past his marker and strode beyond Nemanja Vidic before planting his finish into the corner, courtesy of a slight deflection off the Serb, to end of Moyes' hopes of silverware in his first season.