Tiemoue Bakayoko scored to make it 6-6 on aggregate at the Stade Louis II but the Ligue 1 leaders progressed on the away goals rule, having lost 5-3 at the Etihad Stadium last month.
Kolarov, deployed once again at centre-back by manager Pep Guardiola, battled gamely against a first-half onslaught as Kylian Mbappe and Fabinho fired Monaco into the ascendancy but he was caught out of position when Bakayoko powered home in the 77th minute – six minutes after Leroy Sane's close-range strike appeared to have City bounded for the next round.
Guardiola's side are 10 points adrift of leaders Chelsea in the Premier League and must regroup for a tough domestic assignment against Liverpool on Sunday, while they face Arsenal in the semi-finals of the FA Cup at Wembley next month.
And Kolarov is still optimistic that his side can put their European exit behind them and finish the season on a high, telling the club's official website: "We have two months until the end of the season so I hope we are going to finish in the best way possible in the Premier League and FA Cup.
"We have to work hard to improve every game and we will see where we can reach in the end.
"The aggregate score was 6-6, they scored more goals away and they played very good. I think we are a better team than them, but in the moment we didn’t play the first half like we wanted to. I'd like to congratulate them on going through.
"It's very painful in this moment for us and a difficult day but we have to look forward to the next games."
Right-back Bacary Sagna echoed the thoughts of Guardiola and centre-back John Stones by blaming City's exit on an abject first-half showing.
"It's a massive disappointment. I think we forgot to play during the first half," Sanga told the club website.
"We forgot to fight as a team, defend as a team and press as a team. We gave them too much respect and against these kind of players we put ourselves in danger.
"At half-time we changed the system and I think it worked quite well. We managed to create more chances – unfortunately we didn’t score [enough].
"[Against Liverpool] I expect a reaction from this team. I expect more desire, to show more passion. We did show passion in the second half but it was not enough."