Shakhtar ran out 2-1 winners in Ukraine, coming from behind to win thanks to goals from Facundo Ferreyra and Fred.
"My team played with focus tonight, perhaps without as much quality in the final third, but we knew that sooner or later they would make a mistake and we'd pounce on it", Di Francesco told Mediaset.
Shakhtar's desperation for the away goal they now required left gaps in defence and Ivan Ordets was dismissed for halting a late Roma breakaway, Alisson holding on for a fourth consecutive home Champions League clean sheet to seal the hosts a place in the draw for the last eight. The thing I liked the most was seeing them in the locker room, as this was a victory achieved by men and a group that goes forward together. "I don't have a preferred opponent: they are all strong teams, but we will play to win against anyone". Substitute Wissam Ben Yedder scored twice after coming on in the second half, with Romelu Lukaku pulling a goal back for United.
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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.
'Roma have had the satisfaction this year, despite some hard times, of getting into the quarter-finals of the UEFA Champions League and we need to be proud of this.
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.
"The standing ovation is not for me but for everyone, we are deservedly in the quarter-finals", said Dzeko of the applause which accompanied him off the pitch.