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.
Having trailed 2-1 from the first leg, Roma looked bereft of ideas for much of the contest at the Stadio Olimpico before Dzeko broke the deadlock in the 52nd minute following a tepid first half devoid of goal-mouth action.
Eusebio Di Francesco's side is the second Italian team to advance to the quarter-finals after Juventus, with the draw for the last eight taking place on Friday.
On Tuesday, Roma will welcome Shakhtar Donetsk while Manchester United will file out against Sevilla.
"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.
Dzeko became the first player to score 50 goals in three of Europe's top five leagues this month
"I'm proud of this team because I think we knocked out a very strong team". Kevin Strootman beat the offside trap with a through ball that Dzeko collected just inside the penalty area on the left, and with Andriy Pyatov out to play it, Dzeko calmly finished between the goalkeeper's legs from 16 yards.
"A fantastic result, a fantastic Roma: among the best eight sides in Europe", he posted. - Iven Ordets red card was the second Shakhtar have received in the competition this season - as many as in their previous four Champions League campaigns combined. "If we had finished off the chances we had in the first leg the match would already have been over".
The 31-year-old has now been directly involved in six of Roma's Champions League goals this season.
The Bosnian striker scored the only goal of the second leg to put the Serie A side through on away goals against Shakhtar.
Antonio Conte could make that four with his Chelsea side face Barcelona on Wednesday.
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