And so it proved, despite a resilient effort from the Barclays Premier League side, who equalised through Lukas Podolski having fallen behind to Bastian Schweinsteiger's close-range effort early in the second half.
There was late drama when Lukasz Fabianski saved a stoppage-time penalty from Thomas Muller for what looked a soft decision for a challenge by Laurent Koscielny on Arjen Robben.
However, unlike previous campaigns all is not yet lost for the Gunners this season, as they head to Tottenham on Sunday looking to get themselves back in the title race and can look ahead to an FA Cup semi-final against Wigan at Wembley in April.
The contest got off to a low-key start by both sides, which was understandable given the importance of the first goal in the balance of the tie.
The first opening arrived on 20 minutes when David Alaba darted down the left and fed Franck Ribery, whose floated cross towards the back post was blasted down and over the bar by Robben
Bayern thought they had sewn up the tie on 27 minutes when Javi Martinez fired home a chipped free-kick by Thiago, but the offside flag was swiftly raised.
Arsenal came close to snatching a crucial away goal when Olivier Giroud's header from a corner was plucked out of the air by Manuel Neuer.
Arsenal's club-record signing Mesut Ozil, whose early penalty miss in the first leg proved so costly, was replaced at half-time by Tomas Rosicky, understood to because of a tight hamstring.
Bayern looked to have made the tie safe when Schweinsteiger converted Franck Ribery's cross on 55 minutes.
Arsenal, though, produced a quick response to level when Podolski, the former Bayern striker, crashed the ball in at the near post.
But despite a good spell from the visitors it was not enough to pull off an upset and at least Fabianski's late save avoided what would have been an unjust defeat.
Arsenal captain Thomas Vermaelen returned to the starting XI after an injury to left-back Kieran Gibbs, while German duo Ozil and Podolski were both part of an attacking line-up
However, Arsenal were only able to name six substitutes, with midfielder Ryo Miyaichi part of the travelling group following some late withdrawals, but it was later discovered he was not eligible for the pre-submitted Champions League squad.
Bayern, runaway leaders of the Bundesliga, were looking to avoid a repeat defeat by the English side, who won 2-0 here last season to go out on away goals.
There was no place in the side for forward Toni Kroos - a target for Manchester United - or indeed Muller, who had both scored in London.
Bayern were soon on the offensive, while Arsenal looked to utilise counters through both Santi Cazorla and Ozil, whose early penalty miss in the first leg proved so costly.
Schweinsteiger drilled in a 20-yard effort, which was straight at Fabianski, standing in for Wojciech Szczesny, who had been sent off in the last game.
On 20 minutes Arsenal were carved open as Alaba darted down the left and fed Ribery, who chipped the ball toward the back post but Robben could not provide a finish
German defender Per Mertesacker then slid in to block a dangerous centre from Alaba, who again had found space down the left channel.
Schweinsteiger halted a surging run from Alex Oxlade-Chamberlain, revelling in a central role, as he charged in from the left towards the penalty arc.
Cazorla's free-kick looped off the wall and behind for a corner, from which Giroud tested Neuer with a bullet header.
Koscielny, who missed the FA Cup quarter-final win over Everton because of a hamstring problem, hacked away Robben's goalbound effort after more pressure from the hosts.
Bayern continued to dominate possession, with midfielder Thiago at the centre of the action.
Mario Gotze drilled in a low effort from 20 yards which Fabianski held at the second attempt.
Wenger made a change for the start of the second half, as Ozil, who again largely under performed on another big stage, was replaced by Rosicky.
Former Chelsea winger Robben - who was fouled by Szczesny which saw the Poland keeper sent off at the Emirates Stadium - took a tumble in the penalty area under a challenge from Cazorla
This time, though, Norway referee Svein Oddvar Moen was not interested and told the Dutchman to get up.
Bayern were ahead on 55 minutes when Ribery got away down the left and cut the ball back into the penalty area, where Schweinsteiger arrived ahead of Cazorla to slot past Fabianski.
The Gunners, though, produced a swift response when Podolski shrugged off Bayern captain and Germany team-mate Philipp Lahm on the right side of the penalty area, before darting goalwards and thumping his shot past Neuer at the near post.
The away goal galvanised Arsenal, as Oxlade-Chamberlain and Cazorla continued to see plenty of the ball in the centre.
Wenger went for broke as he sent on 18-year-old German Serge Gnabry, replacing defensive midfielder Mikel Arteta.
Bayern, though, weathered the storm and closed in on the quarter-finals.
Defeat would have been harsh on Arsenal, but it looked to be heading that way when Koscielny was adjudged to have pushed over Robben.
Substitute Muller stepped up to take the spot-kick, which Fabianski saved with his legs and then scrambled to push away from the lunging Muller with the ball spinning invitingly on the goalline.
Source : PA