Having beaten Arsenal 3-1 at the Emirates Stadium in the first leg in London three weeks ago, Bayern had been odds on favourites to finish the Gunners off in the Bavarian capital to go into Friday's quarter-final draw.
But Arsenal took the fight to their German hosts with striker Olivier Giroud giving the Gunners an early lead before centre-back Laurent Koscielny netted with 86 minutes gone to set up a frantic finish.
Bayern held on and the 3-3 aggregate score saw them sneak through.
"We went close, I am very proud of the performance of the team," said Arsenal manager Arsene Wenger.
"I am disappointed for everyone who supports Arsenal that we couldn't get that third goal.