The Bosnian striker's 52nd-minute goal proved enough after a 2-1 first-leg loss in Ukraine, with 10-man Shakhtar failing to find the late strike they needed after Ivan Ordets was sent off late on.
Roma reached the Champions League quarter-finals for the first time in 10 years as Edin Dzeko edged them past Shakhtar Donetsk 1-0 at the Stadio Olimpico on Tuesday and allowed the Italians to win the tie on away goals.
The Bosnian striker's 52nd-minute goal proved enough after a 2-1 first-leg loss in Ukraine, with 10-man Shakhtar failing to find the late strike they needed after Ivan Ordets was sent off late on.
Cengiz Under's goal in the first leg last month ultimately proved crucial for Roma, but Dzeko was the hero on the night and also proved instrumental in Shakhtar defender Ordets being shown a straight red card with 12 minutes left.
That sending-off led to tensions spilling over, with Shakhtar's Facundo Ferreyra picking up a booking for shoving a ballboy over an advertising board.
Eusebio Di Francesco's side are the second Italian team to advance to the quarter-finals after Juventus, with the draw for the last eight taking place on Friday.
"The standing ovation is not for me but for everyone, we are deservedly in the quarter-finals," said Dzeko of the applause which accompanied him off the pitch.
"We managed to do something Roma hadn't in 10 years, so going forward to play against the best makes us all very proud.
"We are among the eight best sides in Europe."
Roma did not concede a goal at home in the group stage and hammered Chelsea 3-0 at the Stadio Olimpico, so the home fans in the crowd of 47,693 would have been confident their side could get the job done.
The Ukrainians dominated early possession though, and there was a scare for the hosts when Alessandro Florenzi nearly nodded a free-kick into his own net.
Ferreyra then came close after Federico Fazio lost the ball just outside his own box, before Roma struck seven minutes into the second half.
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.
Dzeko was then brought down by Ordets who got a straight red card for his troubles, making Shakhtar's task that bit harder.
Roma pounce on a mistake
"We knew that sooner or later they would make a mistake and we'd pounce on it," said Roma coach Di Francesco.
"If we face an opponent with this attitude and determination, we can go far."
"Roma made the most of a single moment," said Shakhtar coach Paulo Fonseca.
"Dzeko, of course, was a key player, but apart from his goal they didn't have a chance, we had possession. If we had
finished off the chances we had in the first leg the match would already have been over."
Roma held on, and after Italy's shock failure to qualify for the World Cup, their progress to the last eight along with Juventus is a boost for the country.
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.
There will also be at least three Italian coaches in the quarter-finals -- Di Francesco, Massimiliano Allegri of Juventus and Vincenzo Montella of Sevilla.
Di Francesco added: "I spoke with Montella a few days ago and we said we'd meet in the final."
Antonio Conte could make that four with his Chelsea side facing Barcelona on Wednesday.