Seeking a third successive win on the road for the first time since November 2023, the visitors were millimeters away from the lead after just five minutes when Romelu Lukaku's header beat Nicola Leali but not the underside of the crossbar.
In response, only a last-ditch interception from Rrahmani prevented Vitinha's effort from finding the back of the net.
However, the opening goal of the match came in the next attack, with David Neres' beautiful delivery into the danger area headed home by Anguissa to give the visitors a fifteen minute lead.
Matters got worse for Genoa eight minutes later when Rrahmani bundled home his first goal of the season through Stanislav Lobotka's inswinging set-piece, putting Napoli in full control of proceedings.
Scott McTominay had a goal disallowed for offside just before half-time, which would have capped an excellent first half for Antonio Conte's visiting side.
Genoa needed a much improved performance in the second period and needed less than six minutes to halve the deficit. Vitinha's mesmerizing wing play allowed the ball to land on the plate, allowing top scorer Andrea Pinamonti to make a touch in the center of the area before sublimely slotting the ball past Alex Meret into the far corner.
Milan Badelj's towering header allowed the Napoli shot-stopper to maintain their slim lead not long after, as the hosts continued their push for an equalizer.
However, despite the one-way traffic, which saw substitute Mario Balotelli hit the post, Genoa were left empty-handed as a fourth win from their last five league matches gives Napoli every chance of being top of the Serie A tree at Christmas should Atalanta lose. Tomorrow to Empoli.
Conversely, Genoa remain two points above the bottom three after once again failing to record their first home win this season (D6, L4).
Flashscore Man of the Match: Amir Rrahmani (Napoli)
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