With Lavezzi out with an ankle injury and Serie A’s player strike called off on Thursday, Slovakia playmaker Hamsik met a Walter Gargano free-kick on 26 minutes in a match that, while initially controlled by Napoli, finished with the hosts - who missed half a dozen good chances to draw level - dominating play.
Napoli went down to 10 men after Michele Pazienza saw red for a late second booking, but they mounted a rearguard action to seal their second 1-0 win in a week.
An open and entertaining clash saw the visitors start brighter as Hamsik forced Eduardo into a first-minute save, while three minutes later Genoa defender Dario Dainelli made Morgan De Sanctis work with a header.
Genoa had a good chance to take a 19th-minute lead when Marco Rossi thrashed the ball over after a super flick-on by Luca Toni, but it was Napoli who drew first blood after a clever free-kick routine.
Gargano’s ball into the six-yard box was a decent one but the real damage was done by a run by Cavani that distracted the home defence, allowing Hamsik to nod home unmarked.
Genoa were denied an equaliser soon after when Toni shrugged off the challenge of Paolo Cannavaro to drill in a teasing cross that Raffaele Palladino would have converted but for the intervention of Dossena.
The second half continued in much the same fashion when Toni laid the ball up for Omar Milanetto to blast over, while Genoa sub Gergely Rudolf fizzed over a brilliant free-kick that Ranocchia headed over.
Genoa kept knocking on Napoli’s door but they could not convert, and Giandomenico Mesto was the next culprit as he dragged a low finish wide of the far post when Toni was awaiting the tap in.
Toni was responsible for a glaring miss of his own when he put the ball past the near post from six yards after a brilliant run down the left by Rudolf.
The worst miss of the night fell to another Genoa player. Rossi was played one-on-one after Toni’s shot hit Christian Maggio, but with time and space he fired wide, erroneously claiming a penalty when Maggio blocked with his chest.
That was one of many rants from the home side, who were convinced that referee Christian Brighi was favouring Napoli when in fact he did nothing of the sort.
Indeed the hosts got the rub of the green when Pazienza, booked earlier for a challenge on Rudolf, was shown a second yellow card when he slid in on Milanetto but appeared to win the ball.
Tempers flared later when Cannavaro received a yellow card for a challenge on Rudolf that Genoa felt merited a red, but - despite a man advantage and a late barrage from the hosts - Napoli held on to maintain their challenge in what is becoming a four-way title race with Milan, Juve and Lazio.
Champions Internazionale lie 10 points off the pace in sixth and are not scheduled to play until January 6 due to their participation in the Club World Cup.