A 1-0 win, after a 2-1 defeat in Kharkiv, took Roma through on away goals as Dzeko scored seven minutes into the second half with the game's first chance of note.
And although the Serie A side were outplayed for long periods at Stadio Olimpico, the Bosnian praised the way his side did not panic before eventually finding the breakthrough.
"It was a great night for me, for Roma and the whole club," Dzeko told UEFA's website. "We deserved this.
"We were patient. We knew that one goal could see us reach the quarter-finals. We scored that goal and then we did everything all together.
"I did [think a second effort was in] but the ball somehow went wide of the goal. That [one that went in] was the golden one.
"I'm proud of this team because I think we knocked out a very strong team."
He added: "Everyone deserves a standing ovation because we all gave absolutely everything, right up until the last minute. We deserve to go through, to be in the quarter-finals."
And Roma legend Francesco Totti also congratulated his former side on Twitter.
"A fantastic result, a fantastic Roma: among the best eight sides in Europe," he posted. "We push forward onto the next round."