The Gunners trailed 3-1 going into the second leg and needed to win by three or more goals against last season's runners-up to move into the final eight but came up short, winning 2-0 at the Allianz Arena but losing on away goals after the two-legged aggregate finished 3-3.
Olivier Giroud gave Arsenal early hope when he swept home Theo Walcott's powerful low cross from the right in the third minute before Laurent Koscielny set up a grandstand finish with a header from Santi Cazorla's corner in the 86th minute.
The Bundesliga leaders were not without their own chances throughout, testing Gunners goalkeeper Lukasz Fabianski on a number of occasions, but could not find the goal that would have eased them through.
Arsene Wenger's men were just the second side to beat Bayern at the Allianz this season but it was not enough as they have to settle for yet another season without any hope of silverware.
2012/13 UEFA Champions - Round of 16 second leg
Allianz Arena, Munich - 68,000