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.
Astronomers offer more expansive view of universe
Duggar sisters unveil Christian dating rules in new book