The Gunners' were looking for a response following their shock FA Cup defeat to Blackburn on Saturday, but they found themselves two-down just after the 20-minute mark at the Emirates Stadium.
And, while Lukas Podolski pulled one back against his former team-mates, the Bavarian giants added a third late on to take a commanding lead back to the Allianz Arena for the second leg.
Arsenal started brightly but fell behind in the seventh minute after a flowing Bayern attack which saw Thomas Muller's cross from the right drop to Toni Kroos on the edge of the area and he fired in his effort off the turf which bounced up and beyond Wojciech Szczesny.
Bayern then took control of proceedings and went further ahead on 21 minutes.
A Kroos corner from the right was headed at goal by Daniel van Buyten and, while Szczesny kept that out, the ball fell kindly for Muller to score from close range.
Bayern continued to dominate for most of the half and Mario Mandzukic should have made it 3-0 on the stroke of half time but he headed a Philipp Lahm cross wide from close range.
Arsenal pulled one back 10 minutes into the second half when they were wrongly awarded a corner by the referee.
The Bayern defence, which had only conceded a penalty in their last 11 games, were all at sea as nobody went to try and cut out Jack Wilshere's flag-kick leaving Podolski with the simple task of heading in from close range.
Bayern went close on 63 minutes when Szczesny saved from Kroos before Javier Martinez headed the resulting corner over the bar.
At the other end substitute Olivier Giroud's first touch saw him fire straight at Manuel Neuer inside the Bayern area.
However, it was all over on 77 minutes when Lahm's cross from the right found Mandzukic at the back post and his close-range effort hit off Bacary Sagna before looping into the net.
Arsenal have a mountain to climb going into the second leg with Arsene Wenger facing an eighth year without a trophy for the Gunners.