The result means Carlo Ancelotti's side are assured of finishing in second place for a second consecutive season as they moved to 73 points, 10 clear of Inter Milan in third with three rounds of matches remaining.
Cagliari remain in 12th place with 40 points, eight clear of the relegation zone and two behind SPAL and Sassuolo.
Napoli were in stuttering form coming into the game, having won one of their last five league matches.
Their struggles looked set to continue when Nicolo Barella’s backheel set up Pavoletti to fire in the opener after 63 minutes.
But the home side cranked up the pressure as Cagliari goalkeeper Alessio Cragno denied Mertens with a superb one-handed save.
However, Cragno was left powerless when the Belgian rose to head in the equaliser from close range with five minutes of regulation time remaining.
Napoli were then awarded a penalty for handball following a Video Assistant Referee (VAR) review deep into stoppage time, resulting in a red card for Cagliari midfielder Artur Ionita for dissent.
Insigne stepped up and showed great composure to dispatch the spot kick and seal the points.
(Reporting by Alasdair Mackenzie; Editing by Ken Ferris)
