But Dzeko rescued the Italian giants with a strike after 52 minutes to bring their quarter final hopes back to life.
"Everyone deserves a standing ovation because we all gave absolutely everything tonight and we fully deserve to be in the quarter-finals", Dzeko was quoted as saying by Sky Sports Italia.
"We are among the eight best sides in Europe".
The Ukrainians dominated early possession though, and there was a scare for the hosts when Alessandro Florenzi almost nodded a free-kick into his own net.
Shakhtar came close to extending their overall advantage when the Roma defender Alessandro Florenzi headed a floated free-kick into the box inches wide of his own far post.
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.
Dzeko was then brought down by Ordets who got a straight red card for his troubles, making Shakhtar's task that bit harder.
"We achieved something that's been missing for 10 years and are now prepared to face anyone in the last eight".
"I cannot judge [Roma] on one game, but I can say they made the most of their only opportunity", Fonseca said. "If we had finished off the chances we had in the first leg the match would already have been over".
"Dzeko, of course, was a key player, but apart from his goal they didn't have a chance, we had possession". In the 2010-2011 season mallenco refereed match of the Champions League between Ajax and Dinamo (2:1), then worked on the matches in the group stage of the Champions League 2012-2013 between "Dynamo" and "Port" (0:0), and in the 2015-2015 season came to Kiev for the match of 1/4 final of the Europa League between "Dnepr" and "Bruges" (1:0).
Di Francesco added: "I spoke with Montella a few days ago and we said we'd meet in the final".