Taylor Hall took a pass from Cammalleri and brought the puck up the right side into the offensive zone on a 2-on-1 break, passed it back to Cammalleri cutting up the middle and he fired it past Predators goalie Pekka Rinne.
Cammalleri started the Devils' comeback from a 4-1 deficit after two as he fired a rebound into an open net 25 seconds into the third.
Andy Greene and Adam Henrique also scored in the third, Hall had a goal and two assists for the Devils, who were 0-4-2 in their previous six road games. Keith Kincaid stopped 38 shots.
Kevin Fiala had two goals and P.K. Subban and Roman Josi also scored for the Predators.