Rafael Benitez's side were held to a disappointing 1-1 draw by Sassuolo at home last time out, but a brace from Goran Pandev ensured they would claim all three points at Stadio Luigi Ferraris.
Napoli have dropped just two points in their opening six games of the campaign, and sit one point above Roma, who have a perfect record after five matches and will host winless Bologna on Sunday.
For Genoa, it was a second successive defeat, and their fourth in six games this season, with Fabio Liverani's side currently occupying 15th place.
Elsewhere, Milan claimed just their second win of the campaign with a 1-0 victory over Sampdoria at the San Siro.
Valter Birsa converted the winner from Robinho's pass in the first minute of the second half to give Massimiliano Allegri's team their first victory in four games.
Sampdoria have now take just two points from their opening six fixtures, with that run of form leaving them in the bottom three.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.