That away tally by Cengiz Under turned out to be the difference for the Italian side with the score knotted 2-2 on aggregate here on Tuesday, allowing the Romans to reach the quarterfinals for the first time in 10 years, Efe news reported.
Both sides were nervous to start Tuesday's clash at Rome's Olimpico stadium and the first half was plagued by offensive miscues at either end of the pitch.
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.
Despite the shortcomings, Shakhtar was entitled to be the more satisfied at half-time, with their 2-1 advantage still intact.
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.
The hosts would have had another goal just three minutes later if Diego Perotti had not squandered a great chance from close range.
Shakhtar shifted into a high press, creating openings for Roma on the counter and on one of those occasions, Ivan Ordets brought down Dzeko 20 yards from goal to earn a straight red card.
Going down to 10 men left Shakhtar in disarray and their frustration showed when Facundo Ferreyra, who scored in the first leg, plowed into a ball boy on the sideline and the youngster fell backwards over an advertising hoarding.
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.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)