Arsene Wenger, who was returning to the club he led to the French title in 1991, looked on as his valiant side almost completed the greatest comeback in Champions League history.
A 2-0 Arsenal victory at a packed Louis II stadium was one goal short after a first-half strike from Olivier Giroud and Aaron Ramsey's goal with 11 minutes left set up a barnstorming finish.
The Londoners ran themselves into the ground as they pressed for the qualifying goal but an inspired performance from Monaco's Croatian 'keeper Danijel Subasic proved decisive.
Ramsey set up the thrilling finale when he scrambled the ball home after fellow substitute Theo Walcott had his shot come back off the post on 79 minutes.
Giroud then had a point-blank header pawed off the line by the acrobatic Subasic as Arsenal threw everything forward in search of the goal that would send them through to the last eight.
Monaco came into the tie with a 9-0 qualifying record in European competition after winning the first leg while Arsenal were trying to break a five year jinx of failing to reach the quarter-finals.
History was not in favour of the Gunners with no previous teams in the Champions League era managing to score more than once away from home in the second leg after losing the first leg and going on to qualify.
After a cagey start by both teams, Arsenal gradually took control of the first-half with Giroud going close with a 14th minute header before Laurent Koscielny glanced his close range effort off the crossbar.
With over 60 per cent possession during the opening 45 minutes, Arsenal finally got the goal their pressure deserved when the dynamic Giroud blasted home his own rebound after Subasic had blocked his initial shot.
There was a moment of controversy when Alexis Sanchez went down under pressure from Fabinho but instead of a penalty to the visitors, the Norwegian referee booked the Chilean for diving.
The Gunners sensed blood and poured forward for the rest of the half with only a goal-clearance from Aymen Abdennour from Welbeck's shot and a desperate Subasic save from Giroud keeping it 1-0 at the break.
The second half provided more of the same although Wenger's side started to fatigue and left gaps open at the back at the expense of pushing numbers forward in search of goals.
Ramsey's goal gave them renewed energy and Giroud almost made it an historic night but the French side, who last reached the quarter-finals in 2004, when they went on to lose the final against Porto, held on in front their nervous fans for a thrilling victory.
Arsenal join Chelsea as the English clubs to be knocked of the tournament with only Manchester City hanging on by a thread as they go to Camp Nou hoping they can overturn a 2-1 deficit against Barcelona on Wednesday.
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