Away goals from Toni Kroos, Thomas Muller and Mario Mandzukic added to the pressure on the already under-fire Arsene Wenger as his side were largely outplayed and were left on the brink of elimination.
A second-half header from former Bayern player Lukas Podolski had given Arsenal, who were embarrassed by Blackburn Rovers in the FA Cup at the weekend, some hope at 2-1 as they threatened a comeback.
But runaway Bundesliga leaders Bayern, unbeaten in 19 games in all competitions, are now very strong favourites to reach the quarter-finals ahead of the second leg in Germany on 13th March.
Wenger must now lift his beleaguered players for the Premier League run-in, where they face crucial games against Tottenham Hotspur and Everton in their bid to salvage something from the campaign with a top-four finish to again play among Europe's elite clubs next season.
The high-octane opening atmosphere from the home support fell flat as Bayern, some 15 points clear at the top of the Bundesliga, grabbed a crucial away goal on seven minutes, when Muller's cross from the right was not dealt with and Kroos crashed the ball off the turf and past Wojciech Szczesny from 18 yards.
It was 2-0 after just 21 minutes when Daniel van Buyten's header at a corner the near post was parried by Szczesny, and Muller stabbed the ball straight back in back into the net from close range.
Arsenal were in danger of being overrun, with few cool heads as first Bacary Sagna and then Mikel Arteta were shown yellow cards for clumsy challenges.
Having left the field to a cacophony of boos from their own beleaguered fans, Arsenal needed to regroup at half-time.
The Gunners suddenly sparked into life on 55 minutes when Podolski, who had a disappointing three-season spell at Bayern, headed in Jack Wilshere's incorrectly-awarded corner at the far post after goalkeeper Manuel Neuer, who had conceded just once in eight matches, had wandered into no-man's land.
Bayern then scrambled what could be a tie-clinching third goal on 77 minutes, with substitute Arjen Robben sidestepping Thomas Vermaelen before releasing Philipp Lahm on the overlap down the right.
His cross flew through the six-yard box, where Mandzukic got ahead of Sagna, the ball looping up off the Bayern striker's shin, over the goalkeeper and into the net.
Some Arsenal fans had seen enough, and headed for the exit, unsure of when they might see Champions League football at the Emirates again.