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.
Dzeko then curled a shot from the edge of the penalty area inches wide in the 62nd minute before the Shakhtar centre-back Ivan Ordets was shown a straight red card in the 79th for hauling down the Bosnian when he was through on goal.
The decisive moment came early in the second half when Kevin Strootman's through ball caught the Shakhtar defence high up the pitch and Džeko burst through before calmly sliding his finish through the legs of Andriy Pyatov.
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It is the first time in 11 years that two Italian teams are in the quarterfinals, with the draw taking place Friday.
The two teams went into the interval level, with the visitors looking the more risky of the two in the opening stages.
'I'm proud of my team who had 60 per cent possession, despite playing with 10 men at the end'.
Roma defender Alessandro Florenzi almost netted an own goal in the 11th minute, but he was lucky to witness his miscued clearance sail inches past the upright.
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But Roma came back from the break with renewed intensity and Dzeko scored in the 51st minute to bring the sides level at 2-2.
A small scuffle between players broke out 10 minutes before full time after Facundo Ferreyra pushed a ballboy over the advertising boards when trying to retrieve the ball from him.
"What I liked best is that we played like men and that's the most important thing if we want to aspire to certain goals and dreams", said Roma coach Eusebio Di Francesco, in his first ever Champions League campaign.
Dzeko was given a standing ovation when he was substituted in the final minute.
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Sisodia said another such meeting would be called and BJP would be invited again. After the meeting, Maken said the discussion was fruitful.
1 - Starting with the 2011 final, Manchester United have only won one of their last eight Champions League knockout games (D3 L4).