The Reds won 2-1 at the Etihad Stadium to reach the last four with a 5-1 aggregate victory.
Barcelona threw the kitchen sink at Roma late on but with Messi and Ousmane Dembele unable to convert their chances, Roma held on to complete their unlikely comeback win.
With a three-goal deficit to make up from the first leg, City needed an early goal to give them hope and they got exactly that, taking the lead in the second minute.
Defender Kostas Manolas scored the crucial goal, heading in at the near post with seven minutes remaining.
He also detailed how the win was just the latest in a succession of games in which Roma upset the applecart in Europe this season including coming out of a group including Chelsea and Atletico Madrid, before overcoming a risky Shakhtar Donetsk.
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He sent two glorious headed chances wide, the first from a vicious 13th-minute Aleksandar Kolarov corner.
Instead, Lahoz chalked off the goal for apparent offside, even though Sane only scored after Liverpool's James Milner touched the ball.
Goal scorer Dzeko, who had been a January target of several clubs including Chelsea, said he was surprised to see Barcelona struggle so mightily, but added that it was Roma's ability to take the game to Ernesto Valverde's team that resulted in their lacklustre display. Their pace up top can change any tie in an instant.
It was Dzeko's sublime centre-forward play that set up the second.
The other two teams in the semi-final will be determined on Wednesday.
As for Barcelona they return to the site of their fourth European Cup triumph, overcoming Manchester United 2-0 in the 2009 final, having maintained their unbeaten streak in this season's La Liga with a comfortable 3-1 victory over Leganes on Saturday.