The Argentina striker opened the scoring early on at Stadio San Paolo with a close-range header and secured victory for Rafael Benitez's side by converting a penalty 15 minutes from time, after Iago Falque had pulled the visitors level.
However, Higuain appeared to be marginally offside for his first goal and Genoa also seemed to have cause for complaint regarding his second, when Napoli were awarded a penalty despite little evidence of a foul.
Napoli moved ahead after just seven minutes, Higuain left with the simple task of heading home after Genoa goalkeeper Mattia Perin had parried a shot from Jose Callejon.
Falque equalised in the 56th minute with a precise left-footed finish into the bottom-right corner from the edge of the box.
Yet Napoli were not to be denied and, after Callejon had hit the crossbar with a deflected strike, the hosts retook the lead in contentious circumstances.
Higuain went to ground as a free-kick from the left was delivered into the box, and referee Giampaolo Calvarese pointed to the spot to leave Genoa's players stunned.
The resulting spot-kick was duly converted, Higuain shooting hard to Perin's left, and Napoli held on thereafter to leapfrog Lazio and Sampdoria.
In Monday's other game, a freak goal from Silvan Widmer enabled Udinese to win 2-1 at Empoli and climb into the top half of the table.
Riccardo Saponara's penalty drew Empoli level prior to the interval after Antonio Di Natale had headed home Allan's cross to give Udinese the lead.
Widmer then scored a fortuitious 60th-minute winner, his apparent cross from the right drifting over goalkeeper Luigi Sepe and finding the net via the far post.
Udinese were subsequently reduced to 10 men when Panagiotis Kone picked up a second yellow card for an elbow on Mirko Valdifiori, but Empoli could not find a way back and remain 16th.
