Indeed, despite needing only a draw to themselves reach the last eight, this was no backs-to-the-wall job for Paulo Fonseca's Shakhtar as it sought to enjoy itself at Stadio Olimpico.
Chances were few and far between, though, and a lack of cutting edge in the final third was ruthlessly punished by Dzeko seven minutes after half-time.
In the first half, Shakhtar managed to keep the ball and played carefully, searching for the ways to penetrate the opponent's defense line.
Taison then had a chance from 25-yards, but his effort glided past Alisson Becker and over the woodwork.
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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.
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.
Ferreyra clashes with the boy.
Here we look at some of the best Opta numbers ahead of the two matches.
The sides have met five times in the Champions League before, with Shakhtar winning on five occasions.
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For the victors a place in the quarter-finals of the Champions League awaits.
Manchester United had David de Gea to thank for a battling 0-0 draw in Sevilla, but will be hoping to spend more time on the front foot at Old Trafford.
Dzeko was then brought down by Ordets who got a straight red card for his troubles, making Shakhtar's task that bit harder.
The judicial Committee of UEFA announced the referees for the second leg of the 1/8 finals of the Champions League between Shakhtar and Roma. - Edin Dzeko has had a hand in 55 per cent of Roma's 11 goals in the Champions League this season (4 goals, 2 assists).
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