The Germany forward put the ball in the net in the third minute for Lazio and the goal was initially awarded but the referee changed his mind when Klose admitted using his hand. Cavani also missed a penalty as Napoli joined Juventus on 13 points at the top of Serie A.
Stephan El Shaaraway scored AC Milan's first home goals of the season in all competitions to give them a 2-0 win over Cagliari and ease the pressure on beleaguered coach Massimiliano Allegri.
It was Palermo's second last-gasp 1-0 defeat in four days since Gian Piero Gasperini replaced Giuseppe Sannino who was fired after only three games.
Lazio's visit to Napoli got off to a stormy start when Klose punched the ball into the net from a corner, leading to a scuffle as the referee awarded the goal.
Napoli took full advantage of the let-off and went ahead when Uruguay forward Cavani scored with a deflected long-range effort in the 19th minute, then beat the offside trap to run on to Paolo Cannavaro's pass and add the second 12 minutes later.
"Klose's gesture deserves a prize," Cannavaro told touchline television reporters at half-time.
Cavani added his third in the 64th minute as Lazio claimed he was offside but missed out on a fourth 11 minutes later when he blazed a dreadful penalty effort high and wide of Federico Marchetti's goal.
Cagliari, the only winless team alongside Palermo, had Daniele Conte sent off after picking up two yellow cards in four minutes in the second half. El Shaaraway ran on to Massimo Ambrosini's pass to add the second, taking Milan up to 10th with six points.
