"It's really satisfying for me because I think we saw a team being born this evening", said Di Francesco.
"Whatever happens in the Champions League is going to be welcomed and we will face it as a great team, you can be sure about this".
The Bosnian forward netted his first Champions League knockout goal and matched his best output in the competition since the 2009-10 campaign with four goals in total.
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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.
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.
Roma did not concede a goal at home in the group stage and hammered Chelsea 3-0 at the Stadio Olimpico, so the home fans in the crowd of 47,693 would have been confident their side could get the job done.
Romelu Lukaku pulled one back for Jose Mourinho's side in the 84th but there was to be no comeback for United, who drew 0-0 in the first leg, and the visitors could have added more. "The team is growing a lot".
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The Ukrainian side were pushing for an equaliser and their Argentine forward was in a rush to get the ball back in play. Ferreyra was shown a yellow card.
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.
"The standing ovation has to be for the whole team, not just for me". "We managed to do something we'd been missing for over 10 years", he said. "I didn't want to leave (in January) precisely because of matches like these, you play all season for matches like these".
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Pique goes down holding his face but there's no intent from the Frenchman as play continues. Chelsea hit the woodwork twice, but were their own worst enemies in defence.