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The Giallorossi secured a second successive Serie A victory thanks to the Italy international's towering finish, the only strike in a game that was light on clear-cut chances.
That was credit to the defensive work of the hosts, however; Paulo Fonseca's side barely giving up a chance to the opposition as they saw out the contest with conviction and professionalism.
When Mancini rose highest to head home Lorenzo Pellegrini's delivery, Genoa could not say they had not been warned. Both Mancini and Chris Smalling had both missed similar chances even before that, as Pellegrini put in consistently reliable balls from set-piece situations.
Having taken control of the game, however, the Giallorossi struggled to drive home that advantage - although substitute Gonzalo Villar hit the post and Borja Mayoral had a strike ruled out for a narrow offside.
Victory was never really in any doubt, however, even if Gianluca Scamacca briefly had Pau Lopez worried with a low drive from distance.
Roma are back in action on Thursday night - they host Shakhtar Donetsk in the first leg of their Europa League last-16 tie.
ROMA 1-0 GENOA
SCORER: Mancini (24')
ROMA (3-4-2-1): Pau Lopez; Mancini, Smalling, Cristante; Karsdorp, Pellegrini, Diawara (Villar), Peres (Spinazzola); Pedro (Perez), El Shaarawy (Mkhitaryan); Mayoral (Fazio)
Unused subs: Mirante, Fuzato, Santon, Reynolds, Pastore
GENOA (3-5-2): Marchetti; Masiello, Radovanovic, Criscito; Ghiglione (Cassata), Strootman (Scamacca), Badelj, Zajc (Melegoni), Zappacosta; Pjaca, Destro
Unused subs: Paleari, Zima, Zapata, Goldaniga, Behrami, Onguenè, Rovella, Cyzborra
Imperium Partners
Centurion Partners
