A 1-1 draw in Manchester last week meant United claimed a shock aggregate lead at the Allianz Arena when Patrice Evra hit a stunning 57th-minute strike.
The goal awoke Bayern from their slumber though, and Mario Mandzukic quickly levelled proceedings.
And Thomas Muller and Arjen Robben also netted as three goals in the space of 17 minutes sealed a 4-2 aggregate victory for Pep Guardiola's side.
Bayern have now reached the semi-finals of Europe's premier club competition for the third season in a row.
The result signals an end to United's hopes of winning a trophy this season – something that has not happened since the 2004-05 campaign.
Mandzukic, Muller, Robben, Franck Ribery and Mario Gotze all started for Bayern in an aggressive line-up but it was the visitors who were able to create the first real chance.
Darren Fletcher, replacing the injured Marouane Fellaini, created it, playing a long ball forward to Wayne Rooney, who controlled on the edge of the penalty area.
The England international was caught in two minds though, and after missing the chance to find the unmarked Shinji Kagawa to his left, his shot on goal was blocked.
United had the ball in the net in the 17th minute, but the goal did not stand with Antonio Valencia drifting into an offside position when converting a Rooney cross.
Bayern eventually recovered from their slow start and as expected, dominated possession, but they were frustrated by their opponents, who were happy to sit back and soak up pressure.
Clear-cut chances were few and far between, with Ribery and Toni Kroos both missing the target before the break, but the game burst into life after Evra's goal.
The left-back ran onto a loose ball and hit a 20-yard left-foot rocket that flew into the top-right corner, an effort that gave Bayern goalkeeper Manuel Neuer no chance.
Bayern's reaction was outstanding, though, as they responded within 22 seconds from the restart through Mandzukic.
Ribery stood up a cross to the back post and the Croatia forward nipped in front of Evra to head past Manchester United goalkeeper David De Gea from six yards.
Bayern took the lead for the first time in the tie just nine minutes later, with Robben given too much time to cross on the right, and his low ball was swept home by Muller, who reacted quicker than Nemanja Vidic.
Robben then sealed the win with 14 minutes left, as he danced across the edge of the penalty area before shooting into the bottom-left corner, with the help of a Vidic deflection.
Substitute Claudio Pizarro had a late chance to extend Bayern's lead even further but his toe-poked effort was easily collected by De Gea.
It mattered little though, as Bayern's Champions League defence gathered more pace.
