Trailing 3-1 from the first leg in London, the visiting Gunners were only able to score two of the three goals that they required to advance.
Sports Mole looks back over the clash to determine if Arsenal deserved even more from the match inside the Allianz Arena.
On target 9
On target 3
Was the result fair?
Bayern had not lost since October and were the strong favourites going into this tie. However, this was no fluke. Arsenal defended really well and made the most of the few chances that they created. They deserved the win tonight, but you'd have to say that Bayern were better over both legs.
From the very first whistle the Germans did not look to be fully with it. That may have had something to do with the fact that they had such a big advantage from the first leg - it was almost as if they didn't know how to play. But for Lukas Fabianski, they may have got a goal or two but they will need to perform much better if they are to win the competition.
It was the sort of performance that hasn't been seen from Arsenal too often this year. They were defensively solid and sat quite a bit deeper than normal. When they were breached, Fabianski was on hand to save them. Despite the two goals, they didn't cause Bayern too many problems. Still, if you had offered Arsenal the win before the match, they probably would have taken it - qualification always looked to be beyond them after all.
Sports Mole's man of the match
Laurent Koscielny: Arsenal have not been known for great defending too often this season but the French international played a big part in tonight's clean sheet. He made numerous blocks and helped restrict Bayern to mainly strikes from long range. His goal almost allowed his team to pull of the unthinkable as well.
There was nothing glaring committed in this encounter, so it may be a little harsh that Toni Kroos has been given this award. Time and again he tried his luck from distance, but more often than not he troubled the supporters more than Fabianski.
In the first half Pavel Kralovec was pretty quiet, but in the second half he was everything that people dislike about officials in the Champions League. Every single foul seemed to be met with a yellow card and it really wasn't that type of game. It seemed as though the Czech Republic referee wanted a slice of the action.
Bayern: This coming Saturday the Bundesliga leaders will travel to take on Bayer Leverkusen as they look to maintain their 20-point advantage.
Arsenal: The Gunners now turn their attention to qualifying for this competition next season when they face Swansea City in the Premier League on Sunday afternoon.