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The Giallorossi made it hard for themselves - going behind twice in the first half - but eventually gathered themselves enough to end their recent run of poor results, as Bryan Cristante scored what would prove to be the winner early in the second half.
Federico Fazio and Justin Kluivert grabbed the other goals for the Eusebio Di Francesco's side, who moved back into the top half of the table as a result.
Genoa made no secret of their confidence in getting a result in the capital before the match, and lived up to their word as they took the lead through Krzystof Piatek - the in-form striker prodding home from close range just minutes into the contest.
Roma eventually equalised through Fazio - who fired home first-time after an inswinging free-kick fell kindly too him - only to go behind again almost immediately, as Hiljemark scored in markedly similar fashion for the Ligurians at the other end.
Discontent was brewing among supporters as a result, at least until Kluivert grabbed his first Serie A goal in brilliant fashion - running onto a Cengiz Under flick near the halfway line before slotting his shot past Ionut Radu.
In the second half both sides continued to have chances - but it was Cristante who took the one that counted, as he scored in a second successive Serie A game with a powerful finish from just inside the box.
The Giallorossi held on to grab the win, and a valuable three points. Now they look ahead to next Saturday, when they will travel to Turin to face Juventus in their final match before Christmas.
Goalscorers: Fazio (31'), Kluivert (45'), Cristante (59'); Piatek (17') Hiljemark (33')
Roma: Olsen; Juan Jesus, Manolas, Fazio; Kolarov, Cristante, Nzonzi, Florenzi; Kluivert, Zaniolo, Cengiz.
Genoa: Radu; Biraschi, Romero, Zukanović; Rômulo, Hiljemark, Sandro, Bessa, Lazović; Piątek, Kouamé
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