City were however, unable to complete the much anticipated comeback as Liverpool hit back in the second half with goals from Mohammed Salah in the 56th minute, and Roberto Firmino putting the nail on the coffin in the 77th minute.
The crowd at the Stadio Olimpico erupted at the final whistle as the Roma substitutes and coaching staff piled on to the pitch to celebrate with the players. Daniele De Rossi did the honors from the penalty spot, firing a blast by ter Stegen, making it 4-3 Barcelona on aggregate.
Yet it was Liverpool whose approach turned out to be ideal in the circumstances as they followed up some staunch first-half defending to level through Salah in the 56th minute and seal the tie through Firmino 13 minutes from fulltime, clinching their third win against City this season.
"Italy needed a European team, and that's what we were tonight".
"We had to be concentrated not to concede a goal".
But three minutes later, the Barcelona goalkeeper had no chance when Manolas headed inside the far post.
"We scored two own goals, there were some dubious situations and we knew the gap wasn't 4-1".
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"The win is hard to describe".
"It's a painful way to go out because no one expected it with the lead we had", he added.
Valverde did not make any changes until it was 3-0 in the 81st minute, bringing off an exhausted captain Andres Iniesta for Andre Gomes, and only throwing on the speedy Ousmane Dembele and forward Paco Alcacer in late desperation.
Striker Dzeko also waxed lyrical about his side's tactics and said he played with more freedom alongside Patrik Schick in attack, with Radja Nainggolan, who missed the first leg with injury, behind them. "I am just happy to see my team in the draw".
It's the first time that Roma has reached the last four since it lost the 1984 final to Liverpool on penalties in its own stadium - and it's the third straight year that Barcelona has been eliminated at this stage.
Although the visitors suddenly mustered a challenge, they could not force a decisive goal and Lionel Messi and co crashed out in scarcely believable circumstances.
Both sides still remain and could set up a repeat in Kiev on 26 May should they progress.
They trail Real Madrid 3-nil heading into the second leg of their quarter final at the Bernabéu.