Pep Guardiola's side were strong favourites heading into the all-English last eight tie, but they suffered a setback by losing the first leg 1-0 away from home.
The stage was set for a thrilling decider at the Etihad Stadium and City found themselves 3-2 up after a frenetic first 21 minutes of action.
Sergio Aguero then put the Manchester outfit ahead on aggregate early in the second half, only for Fernando Llorente to score an all-important third for Spurs 10 minutes later.
Raheem Sterling thought he had sent City through when he fired past Hugo Lloris in stoppage time to send the home crowd into raptures, but a VAR ruling cruelly denied the Englishman and Tottenham advanced on away goals.
Mauricio Pochettino's men have made it all the way through to the final against Liverpool and ahead of the showpiece in Madrid this weekend, Bernardo Silva admits City squandered a great chance to reach the same position.
“Maybe it was the defeat that cost me the most in my career," the 24-year-old told RTP.
“Because I thought this was our year. Because I thought we were ready.
“And so we lost in the quarter-finals against an English team that we know well, and that, in my opinion, we’ve proven to be better than them in recent years. It was hard.”
City successfully defended the Premier League title this season after finishing a single point ahead of Liverpool in the final table.
Guardiola's team also managed to win both the Carabao Cup and FA Cup, completing a historic domestic treble and re-affirming their dominance of English football.
The Citizens will have another opportunity to add the Champions League trophy to their ever-growing collection next season, having already been seeded in pot one for the 2019-20 group stages.