The Premier League side recorded a 2-0 victory on the night but went out on away goals after the tie finished level at 3-3 on aggregate.
Monaco, which won 3-1 in the first leg, was overwhelmed for much of the contest and was fortunate to escape without being dumped out of the competition.
Olivier Giroud's first half strike and Aaron Ramsey's 79th minute effort ensured a tense finale but the visiting side was unable to find a winner.
No team had ever overcome a two-goal first leg deficit away from home in the Champions League and progressed -- but Arsenal's first half display gave it hope.
Giroud, who has often been criticized this season for failing to take his chances, lashed in from close range with nine minutes of the first half remaining.
Ramsey, a substitute, drove the ball into the net after Theo Walcott's effort had struck the post and Monaco's defenders hesitated.
Giroud had a chance late on to cap a dramatic fightback but his effort was saved on the line.
"The best team went through," Arsenal defender Per Mertesacker told ITV.
"Monaco deserved it because they played much better in the first leg. We played well today but it was a massive deficit, and it wasn't enough tonight.
"We came here and tried absolutely everything, and when you look at the game we could have scored more than two.
"Monaco deserve it and that's absolutely fine. They caused a lot of problems away from home. We have to admit that we regret the first game.
"But if we continue to play like that we'll do well in the FA Cup and the league. But play as badly as we did in the first game and you're out of the Champions League. Monaco deserved it."
Arsenal's exit means Manchester City is England's sole representative remaining in the competition.
It faces Barcelona on Wednesday needing to overturn a 2-1 deficit.
Monaco, which which the final in 2004, joins Paris Saint-Germain in the quarterfinals.