Leading 3-1 from the first leg in which they outplayed Arsenal in North London, Bayern were left nervously hanging on at the end after goals from Olivier Giroud and Laurent Koscielny took Arsenal tantalisingly close to an unlikely recovery.
Giroud netted in the third minute to give the Gunners early hope of becoming only the third team ever to advance in the knockout stage after losing the first leg at home.
Bayern never reached anywhere near the level that has seen them dominate this season's Bundesliga and when Koscielny headed in with five minutes left there was a sense of panic in the air.
Arsenal failed to grab the third goal that would have sent them through, however, and their exit means there will be no Champions League quarter-finalist from England for the first time since the 1995/96 season.
The Gunners, who are by no means sure of qualifying for next season's Champions League, returned with their pride restored though after becoming the first English team to win away against Bayern since Norwich City in the 1993/94 season.
"We should be lucky to have gone through," said Bayern CEO Karl-Heinz Rummenigge. "We did not find our rhythm in the first half and had problems in the midfield. But we are through to the next round and that is the most important thing."
Arsenal manager Arsene Wenger said his side's pale display in the first leg had ultimately proved decisive as they lost in the last 16 for the third successive season.
"Overall, we had a great go," he told Sky Sports.
"The regrets we have are from the first game when we conceded a very cheap goal in the last five minutes. If we had lost 2-1 at home you would see how important that goal was.
"It hurts very much but the positives are the performance and the spirit."
Despite a string of key absentees, including former Bayern forward Lukas Podolski and England midfielder Jack Wilshere, Arsenal shot out of the blocks and made good on their promise of an early goal when Theo Walcott cut into the box and slid the ball across for Frenchman Giroud to ram the ball home.
EARLY BLOW
Bayern recovered from the early blow, helped by Arsenal's sloppy passing, and had a chance of their own with Luiz Gustavo firing over the bar from just outside the box after 10 minutes.
Tony Kroos also tried his luck from a distance as a solid Arsenal defence, including left back Kieran Gibbs, worked hard to keep the Bavarians well away from the box.
Bayern injected more pace in the second half and Arjen Robben, replacing injured Franck Ribery, came agonisingly close to an equaliser right after the restart but keeper Lukasz Fabianski stretched to palm his low drive around the post.
Fabianski, starting for Wojciech Szczesny who was left at home because of what coach Arsene Wenger said was mental fatigue, came to the rescue again when the Dutchman raced clear in the box and the keeper blocked his shot at point blank range.
Robben also curled another effort over the bar after jinking inside as Bayern tried to kill the tie but when Koscielny
