Olivier Giroud’s early goal at the Allianz Arena offered the Gunners hope of pulling off an unlikely comeback, and Laurent Koscielny nodded home late on to set up a grandstand finish, but there was to be no miracle in Munich and the quarter-finals of Europe’s premier club competition will be devoid of British representation.
Bayern went into the tie on the back of formidable defensive displays and with an unbeaten home European record against English teams since Norwich won at the Olympic Stadium in the 1993 UEFA Cup.
Arsenal, though, produced the perfect start as they swept into the lead on three minutes. A slip from defender David Alaba allowed Tomas Rosicky to feed Theo Walcott on the right, and his low cross through the six-yard box was smashed into the roof of the net by Giroud.
Bayern are not used to trailing, having conceded just 10 goals in total in the Bundesliga so far, and were soon back on the offensive as Toni Kroos tested Lukasz Fabianski with a 25-yard drive.
Luiz Gustavo, in for the suspended Bastian Schweinsteiger, then fired over after the ball was only half-cleared from a corner.
Arsenal, though, weathered the storm and maintained their own threat on the counter, with Walcott the main outlet down the right and Rosicky seeing more of the ball in a central role.
The Gunners needed to score another two goals to stand any chance of pulling off the unlikeliest of comebacks, but it was Bayern who opened up the second half strongly as Arjen Robben and Kroos both fired just wide from the edge of the box.
There was brief respite when Rosicky was fouled some 25 yards out, but Santi Cazorla hit his free-kick straight into the wall.
With 20 minutes left, Wenger made a double change as Walcott and Aaron Ramsey were replaced by Alex Oxlade-Chamberlain and Gervinho.
The Ivory Coast forward, so often maligned for his inconsistencies, almost produced a moment of magic when playing a one-two with Cazorla in the penalty area, turning his marker to stab the ball across the face of goal and just wide of the far post.
Fabianski produced another good stop from Thomas Muller as Bayern again went in search of an equaliser.
However, it was Arsenal who set up a final flourish as Koscielny headed in from a corner, with a melee following in the goalmouth as goalkeeper Manuel Neuer laid on the ball.