But the visitors' plan was scuppered within the first 10 minutes.
Kylian Mbappé poked home Bernardo Silva's cross and Monaco were 2-0 up on 29 minutes. Thomas Lemar fed the ball into the overlapping run of Benjamin Mendy and the left-back picked out Fabinho who slotted past City goalkeeper Willy Caballero.
At 5-5 on aggregate, Monaco were into the last eight on the away goals rule. But the tone of the tie changed when Leroy Sané scored in the 71st minute to give City a 6-5 overall lead. But that advantage lasted than six minutes.
Tiemoue Bakayoko's superb header proved to be the winner. "It's an immense joy to score in the Champions League. I didn't expect this," he said. "Over two games we deserved to qualify. Now we want to keep going."
"We played an exceptional second half, but the first one we forgot to be there," City boss Pep Guardiola told BT Sport.
"In the second half we were much, much better but it was not enough. It happens and we'll learn, we have a team with not a lot of experience in this competition."
Monaco coach Leonardo Jardim had challenged his side to draw inspiration from the team which eliminated Real Madrid en route to the 2004 Champions League final. Those players recovered from a 4-2 defeat in the first leg of that year's quarter-finals before knocking the Spanards out on away goals.
Jardim said after the match: "Everyone is happy. We've achieved our objective and we played a great match. We were magnificent in the first half. We pressed City all over the pitch and scored two goals. Obviously in the second half, they came back into the match. But overall I think we were the better side over the two games."