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Pandev angled home after fine approach play from Dries Mertens, who drove into the final third before picking out the striker with a precise pass.
Livorno were compact and organised for much of the first period but fell further behind on 26 minutes.
Inler was the player who extended Napoli's advantage with a strike from distance, but Livorno goalkeeper Francesco Bardi was largely responsible, parrying a soft shot into his own net.
Napoli effectively killed off the contest early in the second half through Jose Callejon, who turned home after collecting a slick pass inside from Pandev.
Bardi then went some way to atoning for his early error by parring a free kick from Mertens, with the Azzurri settling for a 4-0 win that moves them up to second in Serie A.
Marek Hamsik helped Napoli add gloss to the scoreline late on, smashing in from close range after a sweeping break started by Lorenzo Insigne.