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 edged to victory in the second leg and reach the quarter-finals despite a dismal display in which Dzeko's victor was scarcely deserved.
Having trailed 2-1 from the first leg, Roma looked bereft of ideas for much of the contest before Dzeko broke the deadlock in the 52nd minute following a tepid first half devoid of goalmouth action.
Shakhtar, who controlled possession throughout the first half, went down a man in the 79th minute when Ivan Ordets was shown a straight red card for taking down Dzeko from behind just outside the penalty area.
"Everyone deserves a standing ovation because we all gave absolutely everything tonight and we fully deserve to be in the quarter-finals", Dzeko was quoted as saying by Sky Sports Italia.
Kevin Strootman sent Dzeko through as Shakhtar tried without success to play the offside trap, and the Bosnian forward prodded a shot through the legs of goalkeeper Andriy Pyatov and into the net for his fourth goal in the Champions League this season.
"We are among the eight best sides in Europe".
The judicial Committee of UEFA announced the referees for the second leg of the 1/8 finals of the Champions League between Shakhtar and Roma.
Romelu Lukaku replied for the Red Devils, but it wasn't enough after the first leg ended goalless in Seville.
15 - Manchester United haven't managed to score more than one goal in any of their last 15 Champions League encounters with Spanish opposition (seven goals in total).
Antonio Conte could make that four with his Chelsea side face Barcelona on Wednesday.