The Italian giants scripted an unlikely comeback from 1-4 deficit of the first leg and secured the semi-final berth after winning 4-4 on away goal advantage.
Manolas, whose own goal helped contribute to the deficit, scored off a glancing header on a corner kick, making it 3-0 on the day and putting Roma ahead via the away goal tiebreaker.
But it was Roma who broke the deadlock after six minutes when De Rossi's pin-point pass was met in the box by Dzeko, who got the ball under his spell with a brilliant first touch before prodding past Ter Stegen from six yards.
Shortly after the half-hour mark, Roma fans protested for a perceived hand ball inside the area but it appeared that Samuel Umtiti used his chest to deflect a shot from Schick. "We so wanted to win this tournament but we've failed to do so again".
"They showed attitude and for whatever reason we didn't know how to respond, or to make chances", Barcelona midfielder Sergio Busquets said.
Later tonight, Bayern, Sevilla, Real Madrid, and Juventus will face off to determine who joins Liverpool and Roma in the semi final stage of this season's champions league. Suddenly, it was game on, and the hosts needed just one goal to go through, as long as they could keep Messi & co. out.
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"This competition is quite fair and I really think a normal final would have been Manchester City against Barcelona, and now they are both out, but there will still be two other really strong teams in there plus Roma and us".
After a game of two own goals in the previous leg that saw Roma go down 4-1 against the Spanish league leaders, Kostas Manolas headed the decisive third goal in the 82nd minute to put his team in first ever semi-finals since the European Cup final in 1984.
Valverde said he was completely responsible for what happened and should take all the blames for the loss.
De Rossi flashed a header wide in the 69th minute, and 10 minutes later it looked like Ter Stegen had saved the away side's skin with an incredible close-range stop from Stephan El Shaarawy. "The only thing I regret was today was a match in which we had a great opportunity and we didn't seize it". Pique then grabbed Dzeko's left arm and tugged the striker to the ground.
"Perhaps the greatest night in Roma's history", tweeted daily Il Fatto Quotidiano. "It's going to be a tough blow".