Arsenal went into the match requiring a four-goal victory to give themselves a chance of progressing through to the last eight, but they got off to a slow start in North London as Bayern threatened to net an early away goal.
After keeping the Gunners on the back foot, David Alaba sent a shot just wide from distance but Arsenal eventually posed a threat in the final third with Olivier Giroud seeing a header deflected wide.
In the 20th minute, the home side netted the early goal which they required as Walcott powered his way through the Bayern defence before sending a thunderous strike from an acute angle into the roof of the net.
Bayern went back onto the offensive but shortly after the half-hour mark, Arsenal felt like they should have been awarded a penalty after Xabi Alonso felled Walcott, but the Spaniard managed to get a touch on the ball.
Three minutes after half time, Arsenal should have netted their second when Aaron Ramsey floated a cross into the box for Giroud, but he sent a free header over the crossbar from 12 yards.
Arsenal needed to score that goal to realistically have any chance of a miracle comeback, and the tie was put to bed just six minutes later as Bayern netted an equaliser.
Koscielny was sent off after the official behind the goal advised the referee to upgrade his initial yellow card to red for a foul on Lewandowski, and the Polish forward made no mistake from the spot.
The man advantage allowed the visitors to dominate the remainder of the match and with 22 minutes left, Bayern began to inflict further damage on Arsenal.
After a poor clearance from David Ospina, Robben eventually found himself behind the home side's defence and he was able to easily place the ball past the goalkeeper from 12 yards.
Arsenal and their supporters were keen to hear the final whistle but Bayern were ruthless as they reached double figures in the tie without much resistance from their English opponents.
Both goals came through Vidal, who firstly clipped the ball over Ospina after being played through before five minutes later, he was able to tap home after being teed up by Costa.
As fans made an early exit, Arsenal managed to see out the remainder of the game without conceding a sixth but they must face up to the failure of not reaching the last eight of this competitive for the seventh successive year.