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.
Despite being a man down, Shakhtar launched a late onslaught and had several risky raids down the left but the home team's defence held on until the final whistle.
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.
Roma opened the scoring seven minutes into the second half when Kevin Strootman played Dzeko through on goal with a defence-splitting pass and he made no mistake with just the keeper to beat, 1-0.
Diego Perotti fired straight at Pyatov as Roma chased a second and Dzeko then nearly struck again with a wonderful curling shot that just faded wide, before Shakhtar redoubled their efforts. - 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.
There have not been two Italian clubs in the quarter-finals of the Champions League since 2006/07 when Roma joined eventual winners AC Milan in the last eight. In the group stage of 2007-2008 Champions League, the Spaniard recorded the Shakhtar's win over Benfica (2:0).
Dzeko was then brought down by Ordets who got a straight red card for his troubles, making Shakhtar's task that bit harder.
And with the clean sheet, Roma goalkeeper Alisson has yet to concede at the Stade Olimpico in the Champions League this season.