Cengiz Under's goal and a string of Alisson saves in a 2-1 away first-leg defeat had kept Roma in the tie, and Eusebio Di Francesco's men edged to victory in the return to reach the quarter-finals despite a dismal display in which Dzeko's victor was scarcely deserved.
Ukrainians Shakhtar had Ivan Ordets sent off after he pulled Dzeko down by the shirt.
Chances were few and far between, though, and a lack of cutting edge in the final third was ruthlessly punished by Dzeko seven minutes after half-time.
"We took our chance to get the decisive goal and then defended all together". Kevin Strootman beat the offside trap with a through ball that Dzeko collected just inside the penalty area on the left, and with Andriy Pyatov out to play it, Dzeko calmly finished between the goalkeeper's legs from 16 yards.
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Alessandro Florenzi came close to turning in an own goal at the 10-minute mark, one that would have been a crippling blow to Roma's comeback hopes.
Kolarov then crucially swept the ball away from Ferreyra's toe inside the Roma box as the sides went in level at the break.
Cengiz Under's away goal looms large, but Facundo Ferreyra and Fred struck in the second half to tilt the balance to Shakhtar, which went on to rout its next two opponents in the league by a combined 8-0 score.
In the 67 minute, Dzeko should have doubled Roma's lead on the night when Diego Perotti played the strike into space, but he fired wide from close range with an empty net at his mercy.
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With just 10 minutes remaining a ball boy on the side of the pitch held onto the ball a little too long for the 26-year-old's liking.
Roma will be returning to the quarterfinals of the Champions League for the first time since 2008, and Di Francesco said that no matter the result of the draw, "I want to keep on dreaming with this team".
"We made a few too many mistakes in the first leg and they made the majority".
- Roma have kept a clean sheet in each of their last four Champions League home games, winning the last three (W3 D1 L0).
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Roma boss Eusebio Di Francesco said his side were waiting for Shakhtar Donetsk to "make a mistake" that would allow them to advance to the Champions League quarterfinals.
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