The German striker equalised shortly after Bastian Schweinsteiger's second-half goal but Arsene Wenger's men could not break down their opponents and exited the Champions League
The holders dominated much of Wednesday's round-of-16 second leg at the Allianz Arena and, despite being unable to secure a deserved win on the night, they progressed 3-1 on aggregate following their first-leg victory in London.
Arsenal beat Bayern 2-0 in the second leg at the same stage of the competition last year but a repeat performance never looked likely and Bastian Schweinsteiger put Pep Guardiola's side in front early in the second half.
Arsene Wenger urged referee Svein Oddvar Moen to give his side a fair chance before the match and the Norwegian official did just that by allowing Lukas Podolski's equaliser to stand after he appeared to push Philipp Lahm shortly after Schweinsteiger's opener.
Lukasz Fabianski, stepping in for the suspended Wojciech Szczesny, saved Thomas Muller's stoppage-time penalty after Laurent Koscielny upended Arjen Robben but the tie was all over by then.
The Bavarian giants progressed on away goals against Wenger's men just over 12 months ago but this was a far more comfortable victory and Bayern were left to dream of becoming the first team since AC Milan in 1990 retain their European crown.
Guardiola had called on his side to be positive and deprive Arsenal of possession and they adhered to his orders in the opening stages, with Schweinsteiger forcing Fabianski into an early save with a long-range drive.
Bayern were dictating the play and Javi Martinez found the back of the net with a volley 26 minutes in but the Spaniard was correctly flagged for offside, then Laurent Koscielny was well positioned to get in the way of the lively Robben's shot.
Arsenal were struggling to contain the hosts by that point and Fabianski had to be alert to keep out Mario Gotze's shot as they weathered the storm to keep Bayern at bay until half-time.
Mesut Ozil was replaced by Tomas Rosicky at the break due to a tight hamstring and Bayern picked up where they left off but Moen rightly waved play when Robben went down theatrically after bursting into the penalty area.
It appeared to be only a matter of time before Guardiola's team went in front and the opening goal duly arrived 10 minutes after the break when Ribery was afforded too much time to pick out Schweinsteiger and the midfielder finished clinically.
Bayern were only in front for two minutes but had every right to feel aggrieved with Podolski's equaliser after their former player appeared to push Germany team-mate Lahm before beating Manuel Neuer at his near post with a vicious left-footed strike.
Mario Mandzukic was then too casual when presented with a great chance to put Bayern back in front and Fabianski was alert to the danger.
Szczesny's deputy then stopped Muller's stoppage-time spot-kick after Koscielny had brought down Robben but that was never going to prove costly as Bayern cruised through.