Bosnian Edin Dzeko scored the only goal at Stadio Olimpico in the 52nd minute, according to Ukrinform.
Shakhtar's Ivan Ordets was shown the red card in the 79th minute.
Shakhtar defeated Roma 2-1 in their first leg played in Kharkiv, but the Italians progressed to the Champions League on the away goals rule.
The draw for the quarterfinal stage of the tournament will be made on Friday, March 16, and matches will be played on April 3-4 and April 10-11.
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