Olivier Giroud stunned the 68,000 sell-out Allianz Arena with a third-minute goal to give the visitors hope of pulling off a Munich miracle.
Despite Bayern dominating for most of the match, Arsenal held firm at the back and Laurent Koscielny's 84th-minute header equalled the scores on aggregate.
This set up a pulsating finale but the Gunners were unable to find the one last goal required to overturn the Germans' away-goals advantage.
The two-legged defeat means that Bayern become the third European powerhouses in three years to knock Arsenal out at the last-16 juncture, following Barcelona in 2011 and AC Milan in 2012.
Reports that Arsene Wenger would concede the match and play a weakened team proved to be misguided, as the Frenchman selected his best available starting XI. He made five changes from the first leg, with injured trio Jack Wilshere, Lukas Podolski and Bacary Sagna replaced by Tomas Rosicky, Giroud and Carl Jenkinson, and regulars Thomas Vermaelen and Wojciech Szczesny axed in favour of Kieran Gibbs and Lukasz Fabianski.
It took just three minutes for Arsenal to give their 3,000 travelling fans hope with a quite splendidly constructed opening goal. A series of cute passes from Santi Cazorla, Aaron Ramsey and Rosicky sent Theo Walcott bursting clear from the right and his slammed cross-shot into the six-yard box was bundled into the net from two yards by Giroud.
Suddenly, the Allianz Arena was quiet. The home side were initially frazzled, almost as if they could not comprehend that their English visitors could be so presumptuous at this German cathedral.
Bayern gradually began to dominate possession, with Javi Martinez, Thomas Muller and Luiz Gustavo all going close to scoring a first-half equaliser, but this was not the same mighty Bayern of the first leg, the team who were so commanding at the Emirates Stadium that many pundits ranked them as favourites to win this season's competition.
It was almost as if the Jupp Heynckes's team did not want to press their foot fully on the accelerator in case they left themselves open to the sucker-punch.
At the end of a first half which Wenger might have concluded went according to plan, his team still needed to score two more goals without conceding but Arsenal struggled to get a foothold in the game after break. Cazorla and Walcott continued to be the trump cards but some of the visitors' midfield passing was wonky and Giroud lacked the velvet touch to hold the ball up and bring others into play.
Bayern continued to exert most of the pressure, with Arjen Robben's weaving runs from the right flank causing particular consternation to the defensive ranks.
Arsenal made a double substitution after 72 minutes, with Alex Oxlade-Chamberlain and Gervinho replacing Ramsey and Walcott, and Wenger's team immediately began to commit more men forward.
Gervinho and Giroud both had half-chances but could not force Manuel Neuer into a save while, at the other end, Fabianski made a string of acrobatic saves.
Koscielny's late header from an Oxlade-Chamberlain corner set up a thrilling finale but Arsenal were exhausted and unable to create any late chances to complete what would have been a remarkable comeback.
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