The two teams were level at 2-2 following the first leg at the Emirates Stadium, but Joshua Kimmich's second-half header proved to be enough to secure a 3-2 aggregate win for Thomas Tuchel's side.
Bayern have therefore progressed to the final four of the competition alongside Borussia Dortmund and Paris Saint-Germain, with Arsenal dropping out at the quarter-final stage.
It is a second straight loss for Mikel Arteta's side, who entered the contest off the back of a 2-0 defeat to Aston Villa in the Premier League on Sunday afternoon.
Arsenal beaten by professional Bayern in Munich
England captain Harry Kane had a half-chance in the fourth minute of the contest, but he turned a Kimmich cross wide of the post, as Arsenal were given an early warning at the Allianz Arena, before the forward struck wide from distance less than two minutes later.
Gabriel Martinelli then struck wide from distance in Arsenal's first opening of the contest, but the opportunities were few and far between in the spell that followed, with both teams enjoying periods of controlled possession.
Gunners goalkeeper David Raya made a smart save to deny the dangerous Jamal Musiala in the 24th minute, before Bayern stopper Manuel Neuer was called into action to keep out Martin Odegaard and Martinelli in quick succession down the other end of the field, with Arsenal looking dangerous.
Neuer also kept out a header from Kai Havertz late in the first period, but a cagey first period finished goalless, with neither team prepared to take too many risks in the opening 45 minutes of the second leg.
Bayern hit the post twice in the opening minute of the second period through a Leon Goretzka header and powerful Raphael Guerreiro strike, with Arsenal coming close to falling behind in the tie.
Arsenal's focus now purely on the Premier League
The home side did make the breakthrough in the 63rd minute of the contest, though, and it came from Kimmich, who powered a header into the back of the net from a Guerreiro cross.
Leroy Sane fired over the crossbar at the far post in the 66th minute, with Bayern looking for a second, and Arsenal were finding it difficult to fashion any serious opportunities.
Raya was called into action late on to deny Musiala, and all of the pressure was coming from the Bundesliga champions, as they looked for a second to kill off any hopes of a late comeback for Arsenal.
There was to be no late drama at the Allianz, though, with Kimmich's flying header proving to be the difference between the two sides in the second leg of the quarter-final.
Arsenal's attention will now switch back to the Premier League, preparing to head to Molineux on Saturday evening for a potentially-tricky affair with Wolverhampton Wanderers.
Bayern, meanwhile, will return to Bundesliga action away to Union Berlin on Saturday, and Tuchel's side are now locked in a battle for second with Stuttgart after Bayer Leverkusen secured the title last weekend.