Shop
Dzeko wrapped up the contest with his goal 15 minutes from time at the Stadio Luigi Ferraris, giving Paulo Fonseca's side a two-goal advantage they would not squander.
First-half strikes from Cengiz Under - an inswinging cross that no-one could get a touch on - and an own goal from Genoa captain Davide Biraschi had given Roma a commanding lead, only for Goran Pandev to get one back on the stroke of half-time.
But Dzeko's calm finish, after Lorenzo Pellegrini had fully capitalised on an error from Mattia Perin to play him in, ensured his side would end up with all three points.
In an entertaining contest at the Marassi there was plenty for both goalkeepers to do, with Pau Lopez making a number of fantastic saves at crucial points for the visitors.
But Fonseca's side always had the edge in the game and created the clearer openings, even if a bit of luck was attached to both of their opening goals.
In the second half, however, it was Dzeko whose calm head in front of goal proved decisive, as the visitors professionally saw things out to pick up their second win in the space of four days.
The Giallorossi are back in action on Wednesday, as they travel to face Juventus in the quarter-finals of the Coppa Italia.
Their next Serie A encounter is on Sunday, as they host Lazio in the Derby della Capitale.
Imperium Partners
Centurion Partners
