The Gunners went down 3-1 in the first-leg at the Emirates but managed to comeback on 3-3 aggregate at the Allianz Arena and exited by account of the away goals rule.
Oliver Giroud stunned the home side with a third-minute goal to give Arsenal an early lead and hope of pulling off a miracle. No other team in the history of the competition had overcome a deficit of three away goals in the second leg.
Bayern dominated most of the match and came close to seal qualification of the quarter-finals, but Arsenal fought back and Laurent Koscielny scored in the 84th minute to raise hopes of a incredible comeback.
Bayern became the third European powerhouse in three years to knock Arsenal out at the last-16 juncture, following Barcelona in 2011 and AC Milan in 2012.
All English sides, champions Chelsea, Manchester City, United and now Arsenal have crashed out of Europe and it is the first time since the 1995-96 season that there is no Premier League side in the last eight.
Malaga beat Porto 2-0
Malaga's fairytale run in their first Champions League season after the Spaniards overturned a 1-0 first-leg defeat by Porto to win the return-leg 2-0 at the Rosadela stadium.
Young playmaker Isco, who won the 'Golden Boy' award as the best U-21 player in Europe last season, had levelled the tie with a fine shot just two minutes from half-time.
Roque Santa Cruz, on loan from Manchester City, scored the crucial goal on 77 minutes from a corner to seal their passage to the quarter-finals.