The striker turned home from close range midway through the first half - thanks in large part to a glorious, clever assist from Stephan El Shaarawy - to decide the game for the Giallorossi, and make it back-to-back 1-0 wins under Claudio Ranieri.
The result will be a further boost for the Giallorossi's hopes of qualifying for next season's Champions League.
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In what was a difficult game for Ranieri, that saw Ivan Marcano and Juan Jesus started at full-back in a somewhat makeshift backline, the Giallorossi weathered a few scares throughout the match - especially in the first half, as Kevin Lasagna and Rolando Mandragora both went close.
In the second half, however, the decision to bring on both Lorenzo Pellegrini and Alessandro Florenzi seemed to open up the match - with Dzeko and Cristante both going close long before the former finally made the decisive breakthrough.
The home side saw the game out expertly, setting up a huge meeting with Inter Milan at San Siro next Saturday.
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