The Predators scored the first goal of the game, but Marc-Andre Fleury had to face just 18 shots and blanked Nashville for the final 57:32.
Neal scored on a deflection with 4:14 left in the opening period to put the Penguins in front. Kris Letang and Sutter added second-period goals while keeping the Penguins atop the Metropolitan Division.
Nashville lost its fourth in a row and finished a grueling road trip with a 2-5 mark. The Predators' lone goal came from Gabriel Bourque.
The four Pittsburgh goals were scored against Marek Mazanec on 25 shots. Mazanec was replaced in the third period by Carter Hutton.