The key point of the fight occurred in the 52nd minute - Dzeko, taking advantage of a positional mistake Ordets, broke out bore down on goal and beat the goalkeeper of the national team of Ukraine.
Shakhtar led 2-1 from the first leg and would have advanced with a draw.
Not since the 2006-07 season had Serie A placed two teams in the final eight, with AC Milan going on to win the competition that season.
"We are among the eight best sides in Europe".
Juve got through to the quarters after beating Tottenham Hotspur 2-1 at Wembley and 4-3 on aggregate. Substitute Wissam Ben Yedder scored twice after coming on in the second half, with Romelu Lukaku pulling a goal back for United.More news: Apple to host surprise education-focused media event on March 27
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The victor against Shakhtar was his fourth in Champions League this season and 18th in total for the club this campaign.
But Dzeko rescued the Italian giants with a strike after 52 minutes as he slotted the ball into the bottom right corner.
That gave Roma confidence and it nearly doubled its lead 10 minutes later but Dzeko fired narrowly wide. "The team is growing a lot".
The boy was pushed over an advertising board by Shakhtar Donetsk player Facundo Ferreira after delaying returning the ball late in Roma's 1-0 win on Tuesday.
The incident resulted in a scuffle between the Roma and Shakhtar players, and Ferreyra received a yellow card.
Dzeko was given a standing ovation when he was substituted in the final minute. "I want to continue to dream with this team". "We managed to do something we'd been missing for over 10 years", he said.