Diego Perotti fired straight at Pyatov as Roma chased a second and Dzeko then nearly struck again with a wonderful curling shot that just faded wide, before Shakhtar redoubled their efforts.
Cengiz Under's strike and a string of Alisson saves had kept Roma in the tie in the 2-1 away first-leg defeat, allowing Eusebio Di Francesco's men to edge to victory in the second leg and reach the quarter-finals despite a dismal display in which Dzeko's victor was scarcely deserved.
A determined AS Roma beat Shakhtar Donetsk 1-0 yesterday to reach the Champions League quarter-finals for the first time since 2008 as a magnificent finish by Bosnia striker Edin Dzeko sent them through on away goals.
Visiting keeper Andriy Pyatov was little troubled by a shot from Dzeko in the first minute and Shakhtar was generally better on the ball, though they could not translate that superiority into a goal. The thing I liked the most was seeing them in the locker room, as this was a victory achieved by men and a group that goes forward together.
'I'm proud of my team who had 60 per cent possession, despite playing with 10 men at the end'. We scored that goal and then we did everything all together.
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Roma hadn't conceded in its three Champions League matches at the Stadio Olimpico this season but almost netted an own goal when Alessandro Florenzi tried to head clear a free kick but sent the ball flashing across the face of goal.
Ordets was shown a red card for tugging on Dzeko's shirt before a scuffle broke out on the pitch after two Shakhtar players pushed a ballboy over the advertising boards for holding on to the ball.
'Roma have had the satisfaction this year, despite some hard times, of getting into the quarter-finals of the UEFA Champions League and we need to be proud of this.
Dzeko was given a standing ovation when he was substituted in the final minute.
Roma hunkered down for the final minutes against a Shakhtar squad that battled to the very last second of the four minutes of added time.