The Bosnian's 76th minute poke put the match beyond reach for the Sicilians, and got Roma back to winning ways after a recent rough patch featuring three defeats spread over the Italian Cup, league and Europa League.
"It hasn't been a great week for us," Dzeko told Premium Sport. "So it was important to win tonight, and not to concede any goals. We'd lost three games, so this will give us back lost confidence."
Juve break Milan's heart in injury time
Palermo are set for relegation and succumbed to early pressure at the Barbera Stadium when Stephan El Shaarawy got on the end of Clement Grenier's superb inswinger on 22 minutes to fire past Andrea Fulignati.
On Grenier's Serie A debut, the Frenchman also earned a caution on the hour and indirectly contributed to Roma's second.
After being replaced by Dzeko in the 64th minute, the Bosnian scored to give the visitors some breathing space.
Conte tells Mourinho to deal with his touchline exuberance
Bruno Peres added the cherry on the cake, capping a counter-attack to slide the ball under Fulignati in added-on time.
Roma's 20th win restores their two-point lead over Napoli, who occupy the third and last Champions League qualifying place with a six-point lead on Inter Milan following a 3-0 win at Crotone.
Days after crashing out the Champions League to Real Madrid, Lorenzo Insigne struck twice, once from the spot, and Dries Mertens added a third as Napoli compounded Crotone's relegation woes to leave them second-from-bottom two points above Pescara.
Pjaca, Alves see Juve past 10-man Porto
But at the San Siro, Inter underlined their top three ambitions with a 7-1 rout of Atalanta that saw Argentinians Mauro Icardi and Ever Banega strike a hat-trick apiece.
Atalanta were confident of causing an upset having beaten Inter, Napoli and Roma at home in a season that is turning out to be their best to date.
Coach Gian Piero Gasperini was humble in defeat, after a dismal display that saw Atalanta drop two places to sixth.
"We got a lot of things wrong today, but in the kind of season that we're having, we can allow ourselves the odd slip-up," he said. "We're still targeting the Europa League."
Icardi completed his hat-trick in nine first-half minutes — including one from the spot when he was tripped in the area — to fire the Nerazzurri to a 3-0 lead by the 26th minute, with Banega netting twice and Remo Freuler hitting a 42nd-minute consolation as Inter took a 5-1 half-time lead.
The hosts twisted the knife seven minutes after the restart when ex-Atalanta midfielder Roberto Gagliardini slammed the ball past stricken goalkeeper Etrit Berisha and into the top corner, before Argentina midfielder Banega completed his hat-trick with a deflected free-kick on 68 minutes.
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