Slovakian midfielder Hamsik opened the scoring on 15 minutes when he controlled an Ezequiel Lavezzi cross and fired into the corner after a defensive slip before Uruguay's Cavani struck home the perfect penalty after Lavezzi had been tripped.
Nilmar had Villarreal's best chances as the visitors pressed but the Brazilian striker left his shooting boots at home, poking wide when clear on goal just before half-time as the Spanish side suffered their second straight defeat.
"We approach every game in the same way and the players are following the game plan to the letter," Napoli coach Walter Mazzarri told Italy's Sky Sport channel.
"There are very few occasions when we enter the pitch not playing as a squad. It's first for the team and then for yourself. Up to now these players have followed me and I'm proud of that."
"We were tired in the second half. [Gokhan] Inler and [Walter] Gargano have run themselves into the ground," said the Italian.
"We couldn't manage at such a high rhythm and we conceded ground. If we only played once a week or had more experienced players like [AC Milan's Alessandro] Nesta or [Clarence] Seedorf, we could do it for 90 minutes."
But fired up by an impassioned 60,000-capacity crowd in the first match in Europe's top competition at the thunderous San Paolo since the days of Diego Maradona in 1991, the home side took charge.
Three minutes later though and the crowd was on its feet after Hamsik broke the offside trap to control Lavezzi's centre before calmly slotting the ball into the far corner.
On 27 minutes, Nilmar shot weakly at Morgan De Sanctis from just inside the area before the Brazilian striker squandered a glorious opportunity five minutes later, toe-poking Cavani's wayward backpass wide of the post with the goal at his mercy.
"It was disappointing to concede two goals early on," Italian Rossi told Sky.
"We played well after that but couldn't find that crucial goal. We need to take the good things out of today's match and see if we can turn things around in the group.
