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.
2014: The Year in Music [PHOTOS]