With Juventus and Napoli expected to duel for this season's Serie A title over the remaining nine games, the focus is on the battle the league's third and last Champions League qualifying spot.
But Roma's eighth consecutive win, coupled with draws for AC Milan and Fiorentina, means they are now just two points behind second-placed Napoli ahead of their trip to Palermo later Sunday, and five points ahead of Champions League rivals Fiorentina and Inter Milan.
Fiorentina, who took a 40th minute lead through Mauro Zarate in Florence, were pegged back by Eros Pisano's late leveller for the league strugglers.
The point took Paulo Sousa's men move back up to fourth place at the expense of Inter Milan, who secured a precious 2-1 win over Bologna on Saturday, but La Viola are now level on points with the Nerazzurri (54).
AC Milan, meanwhile, saw their hopes of finishing among the top three dented after a scoreless draw at Chievo left Sinisa Mihajlovic's men in sixth at 11 points behind Roma.
Dzeko was one of several Roma players slated by coach Luciano Spalletti after a 2-0 reverse to Real Madrid last week left the Italians limping out of the last 16 of the competition 4-0 on aggregate.
Spalletti told Dzeko on Saturday he should be "begging" for his shirt ahead of the trip to the Friuli Stadium.
By the final whistle, the 57-year-old Italian coach had changed his tune.
"Today Dzeko was perfect. He got a lot of unwarranted stick in the press, I hoped he would go out and silence all of them," Spalletti told Rai.
"It wasn't easy to refocus our priorities after elimination from the Champions League. Although today's game looked simple on paper, it was far from easy so compliments to the lads."
Dzeko admitted Spalletti's spur for him to show improvement had ultimately worked: "Every coach has his own approach or way to get you playing. The result speaks for itself.
"I was motivated, because I wanted to win but I'm very happy to have scored."
Although a Napoli defeat would boost Roma's chances of a second-place finish, Dzeko said their focus now is on winning all their remaining games.
"We always go out to try and win, and we'll see where we end up. For the moment, we're happy to be in third place," said the Bosnian.
After Spalletti's call to action, Dzeko responded in positively.
He came close with a glancing header from Stephan El Shaarawy's cross in the opening minutes before breaking the deadlock on the quarter-hour when he ran on to Mohamed Salah's deft touch off El Shaarawy's weighted delivery from midfield to poke the ball past Orestis Karnezis.
Roma twice flirted with conceding an equaliser early after the restart when Wojciech Szczesny parried Cyril Thereau's deflected drive before eeing Duvan Zapata's probing header from a corner come off the upright.
Roma regained control in the 74th minute when Alessandro Florenzi produced a fine touch on the edge of the box to beat Karnezis at his far post with the outside of his boot.
But Roma were forced to endure a nervous finish when Bruno Fernandes pounced on a loose ball inside the area to fire an unstoppable shot past Szczesny and inside the far post five minutes from the end.
Although Fiorentina have been among the top four most of this season, Sousa's men took their winless streak to three games following last week's defeat to Roma and a home draw with Napoli.
Sousa told Sky Sport: "Today I'm sad as we didn't go out and play the way we know how. We let the victory slip away from us."
The draw, however, gave Verona coach Luigi Delneri the belief that his men have what it takes to launch a dramatic fight for top flight survival.
"Securing our survival is still a little way off," said Delneri. "But we showed we have what it takes to take this fight all the way to the finish."