It was always going to be a tall order for Arsene Wenger's men after losing the first leg of their last-16 tie 2-0 in London last month. And so it proved, despite a resilient effort from the visitors, 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 Mueller for what looked a soft decision for a challenge by Laurent Koscielny on Arjen Robben. "Bayern have weaknesses like any other team and when you score anyone can get nervous," said Alex Oxlade-Chamberlain, the Arsenal playmaker who produced an excellent performance.
"When we did that we thought we had a chance but we just couldn't do it. We lacked a bit of quality with the final ball. At the top level you need that. The boys put in a great shift but we can be proud of our efforts."
Arsenal captain Thomas Vermaelen returned to the starting XI after an injury to left-back Kieran Gibbs, while German pair Mesut Oezil and Podolski were both part of an attacking line-up. Bayern, runaway leaders of the Bundesliga, were looking to avoid a second consecutive 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 or indeed Muller, both of whom had scored in London.
Bayern were soon on the offensive, while Arsenal looked to utilise counters through both Santi Cazorla and Oezil. 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.
Arsenal were carved open as David Alaba darted down the left and fed Ribery, who chipped the ball to the back post, but Robben could not 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 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 Olivier Giroud tested Manuel Neuer with a bullet header.
Koscielny, who missed Saturday's FA Cup quarter-final win over Everton because of a hamstring problem, hacked away Robben's goalbound effort after more home pressure.
Bayern continued to dominate possession, with Thiago at the centre of the action. Mario Goetze 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 Oezil, who again largely under-performed on another big stage, was replaced by Tomas Rosicky. Robben - who was fouled by Szczesny which saw the Poland keeper sent off at 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. Arsenal, though, gave 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 the visitors, 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 Mueller stepped up to take the spot kick, which Fabianski saved with his legs and then scrambled to push away from the lunging Mueller with the ball spinning invitingly on the goalline.