The visitors managed to find their passing rhythm early on, although it took until the 20th minute before Dusan Pernis was truly called into action, being forced to keep out Jonathan de Guzman's low drive.
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Duvan Zapata was the next Paretonpei player to come close to making the breakthrough, only for his effort to be kept out thanks to a fine defensive block inside the box.
Napoli, who are currently in eighth place in the Serie A table following a slow start to the season, continued to pile on the pressure with Marek Hamsik firing away a strike which Pernis once more kept out.
The opening goal did come before the interval, though, and it was Hamsik who found it against his former club when he headed home Kalidou Koulibaly's clipped ball into the box.
Dries Mertens skipped past a number of Bratislava players before getting away a shot on the hour as Rafael Benitez's side continued to probe away, and the Belgium international came within inches of finding a way through, only for the ball to ricochet away off the crossbar.
Fifteen minutes from time Gonzalo Higuain jumped highest to extend his team's lead, steering Hamsik's well-placed cross into the box beyond the reach of Pernis.
Napoli managed to hold out for all three points to keep intact their winning start to Group I, while the Young Boys remain bottom following their second successive defeat.