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.
Having gained a two-goal advantage at home, City boss Pep Guardiola had vowed his side would go on the attack in order to finish the job.
But while the Spanish coach can boast the best record of any manager in Europe after 100 games, he opted to start with only Fernandinho in the middle of the park against the aggressive and youthful French side.
Five attack-minded players were deployed in front of the Brazilian midfielder, while Yaya Toure was left on the bench, and it proved a costly move as City were overrun by sharper opponents.
Although they pulled a goal back on the night through Sane – putting them briefly back in front in the tie – the English side never recovered from their poor first-half showing.
Big-money signing John Stones struggled again and Monaco’s winning goal epitomised the fragility of the visitors’ defensive backline, as the impressive Bakayoko was allowed a free header eight yards from goal.
Guardiola has said his maiden City season will be a failure if he cannot deliver a trophy, but barring a dramatic Chelsea collapse in the Premier League, the Spaniard’s only realistic hope of silverware is now the FA Cup.
The City boss made some unwelcome history in France as his side became the first team to be eliminated in a Champions League knockout tie after scoring five goals in the first leg.
The Ligue 1 leaders were missing star striker Radamel Falcao, who had failed so spectacularly in England with loan spells at Manchester United and Chelsea.
But the home side took the game to City, allowing them little time and space on the ball, forcing errors and taking their chances superbly. Although they began to tire in the second half, the 2004 runners-up managed to edge through.
Eighteen-year-old striker Mbappe – who has earned comparisons to retired France great Thierry Henry – found the net after just eight minutes for his 17th goal of the season, fed by the brilliant Portuguese midfielder Silva.
Benjamin Mendy caused all sorts of problems by bombing on from full-back, but man of the match Bakayoko deservedly took his side through with the winning goal.
The towering France Under-21 international controlled the midfield and gained possession nine times – more than any team-mate.