Monaco needed to overturn a two-goal first leg deficit but French prodigy Kylian Mbappe, Fabinho and Tiemoue Bakayoko scored on a memorable European night for the principality club, who underlined their growing status as one of the continent's most exciting sides.
City, who won the first leg 5-3 in Manchester, were completely overwhelmed in the first half and fell two goals behind, but reacted after the break and Leroy Sane reduced the arrears as Monaco seemed to run out of steam.
Bakayoko, however, headed home in the 77th minute to ensure the Ligue 1 club were celebrating at full time after the tie finished 6-6 over two enthralling encounters.
"We deserve this. There is a lot of pride," said Bakayoko, whose goal ensured Monaco are the only French side left in the competition after Paris St Germain were knocked out by Barcelona.
It was the first time City boss Pep Guardiola had been eliminated in the competition's last 16, while City became the first team to exit the Champions League after scoring five goals in the first leg.
City, who have consistently been one of Europe's biggest spending clubs, have now failed to get past the last 16 in four of the last five years.
Meanwhile, Atletico Madrid goalkeeper Jan Oblak was the hero as the Spanish side reached the Champions League quarter-finals for the fourth year in a row after drawing 0-0 with Bayer Leverkusen in an open last-16 second leg on Wednesday.
Slovenia international Oblak made an outstanding triple save to deny Julian Brandt and Kevin Volland in the second half and he made further impressive stops late in the game to shut out the Germans and secure a 4-2 aggregate success.
-The Star