The Gunners could have wished for a better first 45 minutes as Robert Lewandowski’s header, Thomas Muller’s snapshot and David Alaba’s strike put the hosts 3-0 up at halftime. Arjen Robben’s first touch of the game coupled with a Muller breakaway strike came either side of Olivier Giroud’s volley.
A record equalling European defeat leaves Arsenal six points behind Bayern and Olympiakos with two games left.
The Greek side’s 2-1 win over Dinamo Zagreb, achieved via a 90th-minute winner, means that qualification is out of Arsenal’s hands and they are in danger of failing to qualify from the group stage for the first time in 13 years.
Mesut Ozil had a goal ruled out for handball when the score was 1-0 and Arsene Wenger’s team improved after the break.
But Bayern’s relentless pressing and ability to pounce on defensive errors exposed the visitors, who lost for a third time in four Champions League group games.