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.
"We were chasing. We were behind again," Detroit captain Nicklas Lidstrom said of the team's shot output in the third period. "When we're chasing, we're shooting the puck a lot more. That's something we want to do earlier in the game."
Weber gave Nashville a 1-0 lead 3:18 into the game.
With the Predators on a power play, Weber carried the puck from the top of the right circle to the dot, where he sent a slap shot past goalie Joey MacDonald.
"We did enough good things to win," Weber said. "I still think we gave up too many chances, but [Rinne] was really good for us tonight."
The Red Wings had an opportunity to tie it early in the second period when they had a two-man advantage for two minutes after Kostitsyn and Patric Hornqvist were given penalties at 44 seconds.
"The guys that were out there did a great job in terms of the battle," Nashville coach Barry Trotz said of his penalty killers. "We forced them to make a couple of unforced errors, got a save, a big draw and a couple of whistles. Those all count and eliminate that power play."
Just seven of Detroit's 35 shots in the game came in their eight minutes of power play time.
"If you look at the specialty teams, they won the battle," Detroit coach Mike Babcock said. "I thought their penalty killing was outstanding. They put a lot of pressure on us and they got good goaltending."
Kostitsyn's short-handed goal at 14:03 of the second period made it 2-0.
From deep inside the Nashville zone, Ryan Suter sent a long breakout pass to Kostitsyn at the Detroit blue line. Kostitsyn drove toward the net with defenseman Jakub Kindl on him. With just his right hand on his stick, Kostitsyn shoveled the puck toward the net. The puck ski-ramped up the stick of MacDonald and into the net.
"That was a great pass by Suter," Kostitsyn said. "It was almost a breakaway, so I just tried to shoot at the net with one hand. It think it deflected off of the goalie's stick."
With less than three minutes remaining in the third, Kostitsyn scored his second of the game off of a pass from Marcel Goc. It was his 14th goal of the season.
Acquired by Nashville in an offseason trade with Montreal, Kostitsyn has already established career highs in goals and points.
LW Steve Sullivan returned to the Nashville lineup after missing 18 games with a lower-body injury. ... The Predators are 9-1 this season when Weber scores a goal. ... Detroit LW Jan Mursak left the game with an undisclosed lower-body injury and didn't return. ... MacDonald is 0-3 in his career against Nashville.