Bosnian Edin Dzeko scored the only goal at Stadio Olimpico in the 52nd minute, according to Ukrinform.
The decisive moment came early in the second half when Kevin Strootman's through ball caught the Shakhtar defence high up the pitch and Džeko burst through before calmly sliding his finish through the legs of Andriy Pyatov.
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.
Shakhtar went about their business professionally in the opening 45, looking risky on the counter and assured in possession while Roma struggled to generate any chances of note.
"In terms of organisation, I think we are doing much better and we worked on a few set-play routines too".
"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.
Alessandro Florenzi flicked a risky backwards header just wide of his own goal and a nervous home crowd witnessed the visitors maintaining a level of control throughout the first half, Pyatov intercepting when Aleksandar Kolarov led a rare Roma foray.
"A fantastic result, a fantastic Roma: among the best eight sides in Europe", he posted.
- Shakhtar Donetsk failed to register a single shot on target against Roma in this match. In the group stage of 2007-2008 Champions League, the Spaniard recorded the Shakhtar's win over Benfica (2:0). "If we had finished off the chances we had in the first leg the match would already have been over".