An unmarked Matteo Politano broke the deadlock for the visitors after 27 minutes with a low volley inside the box, capitalising on a cross that sailed over the Verona defence.
Two minutes before the break, Kvaratskhelia extended the lead with a precise low shot from a sharp angle, threading the ball between Verona's goalkeeper Lorenzo Montipo and the goalpost.
Ten minutes into the second half, the Georgian winger scored his second, cutting inside a defender and placing the ball inside the far post.
Darko Lazovic got a consolation goal for the home fans with a powerful shot from within the box on the hour mark.
Napoli are in fifth place with 17 points. Verona are 16th with eight points.
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