Shakhtar's desperation for the away goal they now required left gaps in defence and Ivan Ordets was dismissed for halting a late Roma breakaway, Alisson holding on for a fourth consecutive home Champions League clean sheet to seal the hosts a place in the draw for the last eight.
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.
Shakhtar were unlucky not to get an away goal of their own early on when Roma's Alessandro Florenzi nearly headed an in-swinging free-kick into his own net with a glancing clearance. "We could've taken our chances better in the first leg, as if we had done that, we'd already be in the quarterfinals".
Kolarov then crucially swept the ball away from Ferreyra's toe inside the Roma box as the sides went in level at the break.
Shakhtar went about their business professionally in the opening 45, looking unsafe on the counter and assured in possession while Roma struggled to generate any chances of note.
Kevin Strootman sent an imposing ball through to the ex-Man City star, who then effortlessly knocked the strike past the keeper.
Cengiz Under's away goal looms large, but Facundo Ferreyra and Fred struck in the second half to tilt the balance to Shakhtar, which went on to rout its next two opponents in the league by a combined 8-0 score.
The sides have met five times in the Champions League before, with Shakhtar winning on five occasions.
- Shakhtar Donetsk failed to register a single shot on target against Roma in this match.
Roma boss Eusebio Di Francesco said his side were waiting for Shakhtar Donetsk to "make a mistake" that would allow them to advance to the Champions League quarterfinals.
"We made a few too many mistakes in the first leg and they made the majority".
The 31-year-old striker kept his composure and delicately guided the ball into the back of the net with the outside of his right foot, swinging the pendulum in Roma's favour through away goals.