The Gunners were given the runaround in a goalless first half at the Allianz Arena on Tuesday night, with chances few and far between as they sought to turn around Bayern's two-goal advantage from the first leg.
Ten minutes after the restart, Bastian Schweinsteiger scored from Franck Ribery's left-wing cross and although Lukas Podolski quickly smashed in an equaliser, the Gunners could not produce the goals they needed and went out 3-1 on aggregate.
At least they were not beaten, with Lukas Fabianski saving a late penalty from substitute Thomas Muller after Laurent Koscielny fouled Arjen Robben in the box.
Arsene Wenger made three changes to the Arsenal team that started the 4-1 FA Cup win over Everton on Saturday. In came Koscielny, Podolski and Olivier Giroud in place of Kieran Gibbs (ankle), Yaya Sanogo (foot) and Mathieu Flamini, who dropped to the bench. Wojciech Szczesny was suspended after his first-leg red card, so Fabianski was named in goal. The Gunners named only six substitutes; Ryo Miyaichi was part of the travelling group, with Arsenal realising too late that the Japan winger was not in fact eligible for their Champions League squad.
Pep Guardiola made five changes after Bayern beat Wolfsburg 6-1 in the Bundesliga at the weekend. Javi Martinez, Schweinsteiger, Thiago Alcantara, Mario Gotze and Mario Mandzukic replaced Rafinha, Jerome Boateng, Muller and Toni Kroos - who all dropped to the bench - while Xherdan Shaqiri was not involved.
After eight minutes, a great spot from Santi Cazorla enabled him to release the effervescent Alex Oxlade-Chamberlain down the right, only for Dante to blatantly block him off. The Brazil centre-back's booking mean he will be suspended for Bayern's next Champions League match.
Robben volleyed Ribery's cross into the turf and saw the ball go high and well off target, and then Martinez thought he had scored from Thiago's free-kick only to see the offside flag raised.
Just after the half hour, Oxlade-Chamberlain showed impressive dribbling skills as he came in off the left flank before being brought down by Schweinsteiger. Cazorla's free-kick came off the wall for a corner, from which Giroud powered a header at goal - but Manuel Neuer was equal to it.
Tomas Rosicky replaced Mesut Ozil, who had reportedly tweaked a hamstring, at the start of the second half.
Robben wanted a penalty when he went down in the box with Cazorla in close proximity, but Norwegian referee Svein Oddvar Moen was having none of it and told the Dutchman to get to his feet.
However, on 55 minutes, Bayern did take the lead with a simple goal. Ribery dribbled casually into the left side of the area and Bacary Sagna turned away, giving the winger time to play a square pass into the middle where Schweinsteiger finished past Fabianski for his first goal since October.
It appeared all hope was lost but within two minutes, Arsenal equalised. A one-two between Giroud and Podolski put the latter in a race with Lahm down the left and although the Arsenal forward looked to have impeded Bayern's skipper, he was allowed to continue and smash the ball home from a tight angle.
Fabianski saved well from Mandzukic at close range and half chances came and went for both sides, with Serge Gnabry and Flamini brought on by Wenger.
Then, in stoppage time, Koscielny was too physical in his challenge on Robben only for Fabianski to save Muller's resulting spot-kick with his legs before scooping the loose ball away from the goalline.
Arsenal held on for the draw, but ultimately their European adventure came to an end. They now head to Tottenham on Sunday looking to get themselves back in the Premier League title race and can also look ahead to an FA Cup semi-final against Wigan at Wembley in April.