A second-half Bastian Schweinsteiger goal gave Bayern a brief lead before Arsenal's Lukas Podolski equalised with a thunderbolt of a shot as the tie finished 3-1 on aggregate.
"We put in a very, very good first-half and controlled the game by giving away no goal chances," said Bayern captain Philipp Lahm.
"But we made a few too many mistakes in the second-half." Arsenal, who also went out at the same stage of the competition to the same opponents last season, failed to overturn their 2-0 deficit from the first leg three weeks ago.
Amidst hopes of repeating last season's shock 2-0 win in Munich, Arsenal boss Arsene Wenger made six changes from the team which lost to Bayern in London.
Bayern coach Pep Guardiola made three changes from the first leg, switching from a 4-1-4-1 in London to the 4-2-3-1 formation used in last season's European final.
"We didn't gave them many chances, we had expected them to push forward a bit more," said Guardiola.
Bayern got off a bad start when centre-back Dante was booked for a clumsy challenge on Oxlade-Chamberlain after only eight minutes. It was his third of Bayern's European campaign and means he is suspended for the quarter-final first-leg.
Spain's Javi Martinez had a goal disallowed for offside after 27 minutes, while at the other end Bayern's Manuel Neuer saved Olivier Giroud's header.
It stayed goalless at the break even though Bayern enjoyed their customary 70 percent possession.
Off-form Ozil replaced
Wenger made a key change for the second-half as Mesut Özil, who had hamstring trouble, made way for Tomas Rosicky in the attacking midfield role.
It marked a miserable two legs for the off-form Özil, who apologised to Gunners fans via Facebook after failing to convert a key penalty in the first-leg.
"Özil has a hamstring injury and it looks quite serious," revealed Wenger. "He will miss (Sunday's) Tottenham game for sure, he needs a scan. He's out for at least a few weeks."
Soon after the break referee Svein Oddvar Moen ignored Bayern's penalty appeals when Robben went down in the area just before the opening goal.
Ribery found space on the left and crossed inside to Schweinsteiger, playing his first Champions League game since November, to drill home on 55 minutes.
The lead lasted just two minutes as Podolski robbed Lahm and fired home from a tight angle giving Manuel Neuer no chance.
There was drama in added time as Bayern replacement Thomas Mueller had a penalty attempt saved by Fabianski after Robben was fouled by centre-back Laurent Koscielny.