The sides meet again in Rome on March 13.
Shakhtar have been playing their home matches in Kharkiv since the stadium in Donetsk had been damaged by the conflict in Ukraine in 2014.
As the only representative of the eastern part of the continent, Shakhtar Donets face a hard battle this season as they head into Wednesday's Champions League matchup with AS Roma with sparse match practice and lack of quality signings.
Shakhtar were once a regular, fun team to see in the Champions League, putting on entertaining performances and dramatic upsets. AS Roma will nevertheless know much better than to underestimate Shakhtar Donetsk who beat the Italian side in last three encounters back in 2006 and twice in 2011.
Roma player Cengiz Under, left, has been in excellent form this season.
"It was as if we fielded two different teams tonight", said Di Francesco. Dzeko was the architect with a fine through-ball and the Turk took one touch in the box before beating Pyatov, who got something on the ball but couldn't prevent it from crossing the line.
Shakhtar took control and Alisson made superb saves from Marlos and Taison before he was beaten by a brilliant free kick from fellow Brazilian Fred, who sent a looping shot over the wall and in off the crossbar.
The Serie A outfit pushed for an equaliser as the second half continued with substitute Gerson going close with 15 minutes remaining.
Although Shakhtar Donetsk have shown their quality in the Champions League this season, the Miners' impressive record against Roma in recent years should come to an end on Wednesday.
Shakhtar have already come through a tough group which featured Manchester City, Napoli and Feyenoord, and will now be looking to follow that up with a strong showing tonight.