Valverde's side never recovered from that early setback and spent the majority of the game pinned back in their own half, unable to get Lionel Messi into the game.
Despite the visitors commanding a 4-1 advantage from the first-leg, goals from Dzeko, Daniele De Rossi and a dramatic late victor from Kostas Manolas at the Stadio Olimpico saw the Italians advance to the semi-finals due to away goals.
Roma became the third club in the history of the Champions League after Deportivo and Barcelona, who managed to overcome the round of the knockout stage of the tournament after losing their first match with a difference of three goals.
Vehicle horns blared and fans poured into the streets across the Eternal City as supporters hailed an incredible result for a success-starved club.
Roma advance to the last four for the first time since reaching the European Cup final in 1984.
"We all believed, we didn't think about the first leg and unfortunate episodes". "If you come all this way and get to the semi-finals, why shouldn't you believe in getting to the final in Kiev", he asked reporters after the historic win.
Instead the eyes of the world should have been on Rome.
Firmino's 77 minute strike was nail in the cofin for Manchester City as the Reds entered Champions League last 4 with a resounding 5-1 aggregate victory. "Tonight we proved we can play against anyone, as Barcelona are such a strong side".
"It is for moments like this that it is so lovely to live for these colors", recently retired Roma captain Francesco Totti tweeted, adding "Daje Roma!" This squad needs to aim for (the final in) Kiev.
Barcelona return to full strength for the second leg of their Champions League quarter-final with Roma at the Stadio Olimpico in Rome on Tuesday.
They haven't conceded a goal at home in the whole competition and on Tuesday restricted Barcelona to just three shots on target despite the potent attacking firepower boasted by Ernesto Valverde's side.
The local hero was also the architect of the brilliant Edin Dzeko's 6th-minute opener, floating a pinpoint pass that the Bosnian took down expertly and poked past Barcelona keeper Marc-Andre ter Stegen.
Roma forward Edin Dzeko had opened the scoring when he beat the offside trap and latched onto a perfectly weighted through ball from Daniele De Rossi to score in the sixth minute, lifting the hopes of the boisterous home fans in the Olympic Stadium.