Roma were subsequently able to see the game out for an aggregate score of 2-2, which saw them through to the quarter-finals thanks to Under's away goal in the first leg.
Roma have never progressed from any of the four Champions League knockout ties in which they've lost the first leg, which bodes well for Shakhtar.
The Serie A side scored the only goal of the match on 52 minutes when Dzeko beat the offside trap to latch onto a through ball and toe poke the ball past Andriy Pyatov in the Shakhtar net.
"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 a place in the draw for the last eight.
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"We are among the eight best sides in Europe".
'I'm proud of my team who had 60 per cent possession, despite playing with 10 men at the end'.
Roma hadn't conceded in its three Champions League matches at the Stadio Olimpico this season but almost netted an own goal when Alessandro Florenzi tried to head clear a free kick but sent the ball flashing across the face of goal.
The home side came out with renewed goal in the second half and Dzeko delighted their vocal fans when he broke the deadlock with a trademark clinical finish.
A small scuffle between players broke out 10 minutes before full time after Facundo Ferreyra pushed a ballboy over the advertising boards when trying to retrieve the ball from him.
The Ukrainians had last reached the quarter-finals in 2011, which came at the expense of Roma. "Everyone gave their all and we are deservedly in the quarterfinals", Dzeko said.