The Bosnian striker scored the only goal of the second leg to put the Serie A side through on away goals against Shakhtar.
But it wasn't plain sailing for the Italian side, with Paulo Fonseca's side looking the better of the two clubs in the first half.
The visitors came close to increasing their lead from the previous meeting when Alessandro Florenzi almost sliced the ball into his own net.
Taison then had a chance from 25-yards, but his effort glided past Alisson Becker and over the woodwork.
The two teams went into the interval level, with the visitors looking the more dangerous of the two in the opening stages.
But Dzeko rescued the Italian giants with a strike after 52 minutes to bring their quarter final hopes back to life.
Kevin Strootman sent an imposing ball through to the ex-Man City star, who then effortlessly knocked the strike past the keeper.
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The 31-year-old has now been directly involved in six of Roma's Champions League goals this season.
Dzeko came close to doubling his tally just ten minutes later, but his effort drifted wide of the post.
Shakhtar were reduced to ten men after Ivan Ordets recklessly pulled down Dzeko as he was bursting through on goal.
The Serie A side have now reached the last eight of the competition for only the fourth time in the club's history.
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