Bayern Munich lost the battle but won the war to stumble into the Champions League quarter-finals despite a 2-0 home defeat that saw them edge through 3-3 on away goals over the two legs of a last-16 tie that produced more theatre than was expected beforehand.
Arsenal struck with their first attack of the first half on three minutes as Santi Cazorla and Tomas Rosicky combined to some effect.
Rosicky slid a lovely ball into the path of Theo Walcott, who took full advantage of a slip by David Alaba to smash the ball across the face of goal where Olivier Giroud was waiting to net his 15th goal of the season.
Despite being pressed back for large swathes of the match, Bayern were wasteful in possession and in front of goal. They almost paid a heavy price when Laurent Koscielny rose highest in the home box to nod Cazorla's corner into the corner of the net on 85 minutes beyond goalkeeper Manuel Neuer.
With a stunned mood drifting over the Allianz Arena, Arsenal knew one more goal would carry them through to the last eight, but could not find the quality needed to get it as a Mikel Arteta free-kick drifted out of play late on to signal the end of Arsenal's European campaign for another season.
Arsenal are only the third side to register a win over Bayern this term, but in the cold light of day glorious failure means little.
Arsenal face an eighth year without winning any silverware under Arsene Wenger, no matter how you dress it up with the platitudes that will rightly come their way for this win.
Bayern were without a host of first-team players, but were expected to progress. Bastian Schweinsteiger and Jerome Boateng were suspended, with Franck Ribery and Holger Badstuber both injured.
Brazil midfielder Luiz Gustavo filled in for Schweinsteiger while Dutch star Arjen Robben replaced Ribery. Dante played alongside Daniel van Buyten in defence.
Arsenal midfielder Jack Wilshere has been ruled out for three weeks with an ankle injury, while former Bayern striker Lukas Podolski is also out with a similar problem.
Tomas Rosicky returned in midfield while Giroud started. Nacho Monreal was cup-tied, and there was no starting place for Thomas Vermaelen. Goalkeeper Wojciech Szczesny was rested.
Bayern were a massive disappointment throughout the evening, a couple of gears under the levels that saw them ride roughshod over Arsenal with a 3-1 win in the first leg in London three weeks ago.
This was a side unbeaten in 23 matches stretching back to October, but despite being 20 points clear of Borussia Dortmund at the top of the Bundesliga, their listlessness left the home fans giving off the impression they had drowned in a vat of Mogadon.
Toni Kroos did produce a shot at goal from distance on 25 minutes that was held by the recalled Arsenal goalkeeper Lukasz Fabianski as he dived low to his left to make the stop.
Thomas Mueller also drilled a shot into the side of a post from an acute angle after Per Mertesacker made a mess of a clearance, but that was about the home side's lot in the first half.
Arsenal had improved their play after a worrying period of needlessly shedding possession and could have been in for the second goal when Walcott clipped a lovely ball into the heart of the opposition box.
Giroud was again lurking, but could not throw himself to the ball in time to convert a cross that was waiting to be buried on 31 minutes.
Bayern tried desperately to inject some life into their play after half-time with Robben, Kroos, Gustavo and Philipp Lahm all missing the target from distance after 20 minutes of the second period.
Bayern finally managed to penetrate Arsenal on 69 minutes when a Muller backheel sent Robben through on goal, but the Dutch forward seemed to be perturbed by the attention of the excellent Carl Jenkinson as his shot was comfortable enough for Fabianski to beat away.
Gibbs was then needed to make a strong block on Javi Martinez before Fabianksi stood tall to repel a Muller shot.
Arsenal substitute Gervinho - on for Walcott with 18 minutes remaining - almost found a route to goal late on as he prodded wide after some neat close control.
Despite Koscielny's header and more silence in the stadium, Arsenal ran out of time in pursuing the third goal they craved to leave England without a representative in the Champions League quarter-finals for the first time since 1996.
It is a relieved Bayern who take their place in Friday's draw, but they will be aware of the need to perform better than this.
MAN OF THE MATCH
Carl Jenkinson (Arsenal) - Has been berated by some critics, but this was a performance of maturity and no little quality as the lively full-back stood up to his task in defence with some authority.
BAYERN MUNICH: Neuer 6; Lahm 7, Van Buyten 5, Dante 5, Alaba 5; Martínez 6, Gustavo 6; Müller 6, Kroos 6, Robben 6; Mandzuki 6; Subs:Tymoshchuk 6, Gomez 5.
ARSENAL: Fabianski 7, Jenkinson 8, Mertesacker 7, Koscielny 7, Gibbs 7, Arteta 6, Ramsey 6, Rosicky 6, Walcott 7, Cazorla 7, Giroud 7; Subs: Oxlade-Chamberlain 6, Gervinho 7.