The English side were 13 minutes from a place in the quarter-finals after clawing themselves back into a second leg their hosts had dominated, but slack marking from a set-piece allowed Tiemoue Bakayoko to head home the decisive goal.
Having won the competition twice in his time at Barcelona, this is the first time in manager Pep Guardiola's career that he has gone out at this stage.
Monaco lost 5-3 in an extraordinary first leg in Manchester but dominated the first half at the Stade Louis II and opened the scoring through the excellent Kylian Mbappe's poked finish from close range.
The Ligue 1 side, who had scored 123 goals so far this season, deservedly doubled their advantage on the night, punishing City's sluggish start through Fabinho's crisp strike.
City failed to muster any sort of shot in the opening 45 minutes and it took until the 65th minute for Sergio Aguero to call goalkeeper Danijel Subasic into a sharp save.
They forced their way into the game - and back into the aggregate lead - as Leroy Sane swept in when Subasic parried Raheem Sterling's low strike, but their defence could not hold out.
The result leaves Premier League champions Leicester City as the only English team in the last eight.
Monaco join the Foxes, holders Real Madrid, last year's runners-up Atletico Madrid, Barcelona, Bayern Munich, Borussia Dortmund and Juventus in Friday's draw.
"I tried to convince them in all the meetings we had to come here, try to attack and score," said Guardiola.
"My mistake was being not able to convince them to do that."