Kostitsyn's first goal put the Predators ahead for good with 5 minutes left in the first period, breaking a 1-1 tie.
Shortly after Matt Halischuk had given Nashville a 3-1 advantage early in the second, Kostitsyn struck again, gathering a loose puck at center ice, skating in alone on Miikka Kiprusoff and putting it between his pads.
Kostitsyn added an empty-netter and Kevin Klein also scored for Nashville, which won its third straight.
Pekka Rinne stopped 17 shots to earn the victory.
Kiprusoff finished with 22 saves.
Moore to attend retreat in to avoid Kutcher's wedding
Yosemite climber falls 30 feet, suffers major injuries