The visitors took the lead inside the first minute when Emil Hallfredsson latched on to a clearance on the edge of the box and powered a volley into the far corner of the net.
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Hallfredsson almost scored his second in the ninth minute when he burst into the box and had his shot parried by the keeper, before Luca Toni missed the chance to score from the rebound.
The match was being played at a frantic pace and Higuain was the next player to be denied as his powerful shot was turned behind.
Jorginho then tried his luck with a strike from just inside the box that was beaten away by the keeper.
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After the break, Callejon was close to giving his side the lead, but after latching on to Insigne's ball over the top, he ended up lifting his strike over the bar.
Just before the hour mark the hosts scored their second of the match through Hamsik, who saw his initial strike saved, but made no mistake to fire home the rebound.
Nicolas Lopez soon levelled the scores when he dribbled into the box, before firing in a low strike that beat Rafael at his near post.
It took Napoli less than a minute to retake the lead as Higuain drove through the Verona defence and calmly slotted an effort past the keeper.
In the 76th minute the hosts grabbed their fourth of the evening when Callejon was slipped in by Hamsik following a quick break and fired a low strike into the far corner.
Panagiotis Tachtsidis then threatened to pull a goal back for the visitors with a free kick from range that had to be helped over by Rafael.
With five minutes left to play Higuain scored Napoli's fifth of the evening when he slotted home Raul Albiol's pass from close range.
Higuain was denied the chance of a hat-trick when his shot was incorrectly flagged offside and soon after Callejon fired a strike against the post.
The Argentine did manage to get his third of the match in the final minute, however, when he converted from the penalty spot following Hallfredsson's trip on Dries Mertens to complete the scoring.