The Bosnian striker netted the winner at the Stadio Olimpico on Tuesday as Eusebio Di Francesco's side prevailed on away goals after a 2-1 defeat in the opening leg.
Dzeko, who scored twice against Chelsea in the group-stage clash at Stamford Bridge, was heavily linked with the Blues in January as Antonio Conte sought a physical presence up front to lighten the load on Alvaro Morata.
The defending Premier League champions ultimately brought in Olivier Giroud from Arsenal and, with Chelsea facing a daunting trip to Barcelona on the back of a 1-1 draw in their last-16 tie while Roma await the quarter-final draw, Dzeko is pleased he decided to stay put.
"I did not want to go away for this game, all the season is played for games like that," he told Mediaset Premium after Roma joined Juventus in the last eight to give Serie A two Champions League quarter-finalists for the first time in over a decade.
"We have done something that has not happened for 10 years, and being with the strongest is a source of pride.
"We play with everyone, we are among the eight strongest, so we are strong."
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."