Trailing 3-1 from the first leg in London, the Gunners had to win by three clear goals to qualify for the next round outright.
As it was, Arsene Wenger's side almost pulled off the unthinkable. An early strike from Olivier Giroud provided the visitors with belief and they set up a grandstand finish when Laurent Koscielny scored a late header.
It wasn't to be though and the hosting Germans held out to book their place in the last eight of the tournament.
To find out how the match unfolded, you can read our minute-by-minute commentary below.
"It is not mission impossible," Wenger told the club's official website. "It can be done, but first of all you need 100% commitment, attitude and things to go your way. But usually football goes your way when you have the right attitude.
"There is a kind of [feeling of] 'let's really play with freedom'. Let's start strong. We have to create doubt in their mind and you only do that if you have a real go."
"BREAKING NEWS: 2nd Pope elected since Arsenal last won a trophy."
"We had a very good day against Arsenal [in the first leg]; the first half was excellent, the second quite good," he told reporters. "But Arsenal are still Arsenal. For many years they have played great football. I remember [last season] Arsenal lost the first leg 4-0 to Milan but then won 3-0 at home, nearly scoring a fourth goal.
"We are not paying attention to the hype around us. People are saying we are the best team in Europe; that does not affect us.
"There are so many strong teams in Europe and to reach the final you always need a little luck. So my team remains calm and down to earth. We have a long way to go."
"Arsenal have only kept one clean sheet in their last 23 Champions League games away from home."
"We will see if Bayern are too complacent. That could be Arsenal's only chance. No game starts with zero chances. You never know."
There was heck of a pause before he answered the question - almost like he didn't truly believe what he was saying!
"Arsenal should go into the game thinking they can do this. The overriding thing is that they can't afford to concede a goal. A clean sheet gives them a chance.
"Given Bayern's form this season and looking where they are in the German league it looks unlikely Arsenal can get a result, but that being said football is a funny business and anything can happen.
"On their night Arsenal is a match for anybody and can beat anybody. If they turn up tonight it might be an interesting game, but if they concede, that kills the tie."
"En route to the ground and come on arsenal!!!"
"Giroud's third minute goal was the fastest Champions League goal for Arsenal since Freddie Ljungberg scored in the second minute against Ajax in Sep 2005."
Rio Ferdinand: "Go on Arsenal! Have a go at this mob!"
Gary Lineker: "Ooh la la! Giroud gives Arsenal the lead..."
"The reason for the lack of atmosphere is the fact that the active supporters can't get behind the goal in ucl - not the score line.
"Same thing when they thrashed Lille or in any other UCL game this season - in case u watch sky uk and wonder."
"The crowd in the Allianz are a little quieter than they were an hour ago. I'll just enjoy it for now, I think... #hopeisthekiller."