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.