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.
The Spanish outfit progressed to the quarter-finals for the first time in their history, with goals from substitute Wissam Ben Yedder in the 74th and 78th minutes.
Shakhtar's chances of producing a comeback were dealt a huge blow in the 79 minute when they were reduced to 10 men following the sending off of Ivan Ordets after straight red card.
"We are among the eight best sides in Europe, so that means we're strong too".
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Hunter suffered a broken left wrist during the Cavaliers' run at the conference tournament in Brooklyn last week. No. 1 overall seed Virginia will be without one of its star players for the duration of the NCAA Tournament.
The 31-year-old has now been directly involved in six of Roma's Champions League goals this season.
Ferreyra then came close after Federico Fazio lost the ball just outside his own box, before Roma struck seven minutes into the second half.
Neither side could find the breakthrough by half time, but the Giallorossi went on to take the lead not long after the restart when Dzeko sprung the offside trap to latch on to a Kevin Strootman through-ball. "We deserve to go through, to be in the quarter-finals".
"Roma made the most of a single moment", said Shakhtar coach Paulo Fonseca.
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The Bosnian striker scored the only goal of the second leg to put the Serie A side through on away goals against Shakhtar.
Despite being a man down Shakhtar pushed forward in search of the goal that would send them into the quarterfinals of the competition but they were unable to do so, losing 1-0 on the night and exiting the Champions League on the away goals after a 2-2 aggregate scoreline. - This is the first time Roma have progressed from a knockout tie in which they have lost the first leg; ending a run of five consecutive knockout tie defeats.
Antonio Conte could make that four with his Chelsea side face Barcelona on Wednesday.
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Nicol said on ESPN FC when asked who is the best behind Guardiola: "Obviously I would go with Klopp. That didn't happen, but everyone can see that the way is positive at least.