The Partenopei enjoyed a storming first half, scoring four times, but went missing somewhat in the second half as the Serie A strugglers bagged a couple of goals too
Duvan Zapata opened the scoring early on before Jose Callejon - provider for the opener - doubled the scoring with a simple tap-in following a defensive mix-up.
Henrique added a touch of class to proceedings in the 40th minute, thundering in from an absurd angle, before Zapata quickly added his second of the night as the hosts' defence and confidence crumbled.
The Sicilians plucked a goal back via Fabian Monzon eight minutes after the break before defender Norbert Gyomber headed in with quarter of an hour left on the clock, but Napoli held firm for the away triumph.
The win keeps Rafa Benitez's men six point behind second-placed Roma in Serie A, while Catania remain rooted to the foot of the Italian top flight with just eight games left in the season.
Star striker Gonzalo Higuain was rested for the visitors, with Zapata starting in his place, as the Partenopei searched for a return to form after a Europa League exit and weekend loss to fourth-place Fiorentina had left them out of form and at risk of their season derailing.
Napoli started with more possession and Callejon's cross for Lorenzo Insigne ended with the Italy international testing Mariano Andujar - on loan at Rossazzurri from the Stadio San Paolo - in the opening minutes.
It took 17 minutes for the visitors to put the ball in the back of the net when Callejon's swift, low cross was met by the unmarked Zapata at the back post to score the opener.
The home team reacted excellently, though: Monzon cut inside and smashed the ball against the crossbar from inside the box, but Keko's acrobatic attempt flew sideways and Gino Peruzzi's header at close range was tame enough for Pepe Reina to hold on to.
But Rolando Maran's side stitched themselves up for the second goal as Callejon beat the offside trap, watched as the goalkeeper collided with team-mate Nicola Legrottaglie then the ex-Real Madrid man rolled the ball into the net.
Barely 90 seconds later, with the half-hour mark looming, Barrientos inexplicably skied his shot from seven yards out and his team remained two goals behind as their hopes of a result were fast fading.
Napoli eased off the pressure until a few minutes before half-time when Henrique struck brilliantly from a tight angle, lobbing over Andujar then dipping into the top corner for his debut goal for Napoli.
Three minutes later Zapata was threaded into the box, took a second to compose himself and then curled calmly past the Catania goalkeeper to celebrate his second of the night at the Stadio Angelo Massimino.
Maran's men came out after half-time looking a different team and quickly pulled a goal back as a Jaroslav Plasil shot - saved by Reina - fell straight to the feet of Monzon who converted to restore a smidgen of pride to the battered Catania.
The head of steam evaporated after 20 minutes though and Napoli remembered how to play football again, keeping possession from the frustrated hosts in between firing hopeful and half-hearted shots on the Rossazzurri goal.
With 15 minutes to go, however, Catania's Gyomber rose higher than his marker to propel home with his forehead and set up a slightly nervy finale for the Partenopei.
The hosts valiantly pushed for more goals late in the game but were unable to fashion any more clear-cut chances, meaning Napoli could coast to their first competitive victory away to Catania at the 13th attempt.