While Napoli posed a challenge throughout the match, Inter were lethal in front of goal with Hakan Calhanoglu, Nicolo Barella and Marcus Thuram all scoring at crucial moments when Napoli were gaining momentum in the match.
Inter lead Serie A with 35 points, two points ahead of second-placed Juventus after 14 matches.
"Coming to Naples you have to expect some difficulties. Inter manager Simone Inzaghi told DAZN. "We had prepared in a certain way to push our wing-backs against their full-backs because the further away you keep Napoli from the penalty area the better.
"The only time we really struggled was when we were defending too deep. Winning like this in Naples is satisfying, but we also know it is only Week 14 and there is a very long way to go."
The loss is another setback for Napoli's hopes of retaining the title as they now trail Inter by 11 points. They are in fifth spot level on 24 points with AS Roma who are fourth.
Despite wearing their orange third kit, Inter's players had trouble finding each other early in the match. Their momentum frequently faltered and Napoli created opening down the flanks.
Inter then scored against the run of play just before halftime as Calhanoglu fired a precise low shot from outside the box into the bottom corner of the net for the breakthrough.
In the second half, buoyed by a noisy home crowd, Napoli continued their offensive push, with Kvaratskhelia's angled drive on the hour mark narrowly missing the target.
But in the 61st minute Napoli's second-half momentum was derailed when Barella received a cross in the box and skilfully slalomed past two defenders before placing the ball beyond keeper Alex Meret to double Inter's lead.
Thuram sealed the points with a goal five minutes from time when he ghosted into the box unmarked to tap home substitute Juan Cuadrado's precise cross for Inter's third.