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.
The visitors' hopes of snatching the equaliser they needed to progress were hampered when Ivan Ordets was sent off for a 79th-minute professional foul on Džeko.
A momentary loss of concentration ensured the Pitmen conceded a goal that knocked them out, with Ordets' red card not helping their cause.
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.
He added: "Everyone deserves a standing ovation because we all gave absolutely everything, right up until the last minute. All the other teams are strong but we are strong too".
"A fantastic result, a fantastic Roma: among the best eight sides in Europe", he posted.
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"I cannot judge [Roma] on one game, but I can say they made the most of their only opportunity", Fonseca said.
Goal scorer Dzeko had no doubts, saying Roma were "deservedly" through, and he said the chance to move on with Roma was a big reason why he stayed on in January, despite interest from Chelsea.
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.
Chances were few and far between, though, and a lack of cutting edge in the final third was ruthlessly punished by Dzeko seven minutes after half-time.
"If we face any opponent with this attitude and determination, we can go far".
- 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. In the group stage of 2007-2008 Champions League, the Spaniard recorded the Shakhtar's win over Benfica (2:0).