Kostitsyn scored twice and Pekka Rinne stopped 35 shots and the Predators beat the Detroit Red Wings 3-0.
Captain Shea Weber added a goal for Nashville, which snapped a four-game losing streak.
Including a 3-0 home loss to Columbus on Friday, the Red Wings have been shut out in consecutive games.
The second-place Predators are five points behind the Central-leading Red Wings.
Rinne was stellar in earning his third shutout of the season and 17th of his NHL career. After turning aside six Detroit shots in the first, he saw 12 shots in the second and 17 in the third.