A dominant Bayern, leading 2-0 from the first leg, had a Javi Martinez volley ruled out for offside before the break.
Bastian Schweinsteiger gave Bayern the lead from close range before Lukas Podolski shot high into the net to level the score on the night.
Thomas Muller had a late penalty saved as Bayern won 3-1 on aggregate.
Arsenal's European exit leaves them just two more chances to end a nine-year trophy drought this season. Arsene Wenger's men are in the FA Cup semi-final and seven points behind Premier League leaders Chelsea.
The damage was done in the first leg at Emirates Stadium, where Arsenal will rue an early missed penalty by Mesut Ozil and the dismissal of goalkeeper Wojciech Szczesny.
Few gave the Gunners much chance of overturning the 2-0 deficit at the defending European and German champions, who won a record 16th Bundesliga game in a row against Wolfsburg on Saturday to go 20 points clear at the top.
No side has ever retained the Champions League crown but, unbeaten in the league for 49 matches, Bayern are strongly fancied by many to make history in Lisbon in May.
Arsenal did go into the match with reasons for hope, having been the last team to prevent the German side from scoring when theybeat them 2-0 at the Allianz Arena at the same stage last season, although the Gunners went out that night on away goals.
Bayern's defeat by Arsenal last season was the only game the Bavarians had failed to score in in their last 100 matches and a composed start by the hosts made the prospect of another shut-out an unlikely one.
(BBC Sports)