The visitors had to play the last 25 minutes with 10 men after Christian Maggio received a second yellow card, having picked up his first booking early on in the game.
Callejon broke the deadlock midway through the first 45 minutes when a inch-perfect pass from Marek Hamsik put the Spaniard through on goal, and he finished coolly with only the keeper to beat.
Benitez's side extended their lead and ultimately sealed the win when Gabbiadini curled his left-footed effort past Brkic from 20 yards out to make it 3-0.
The result keeps Napoli in fourth, trailing Roma, who sit in the last Champions League spot, by five points.
The hosts remain in the relegation zone, nine points from safety with only seven games left.