The Predators moved within four points of the Red Wings in the battle for home-ice advantage in the first round of the playoffs.
Nashville's Patrick Hornqvist and Detroit's Drew Miller scored 13 seconds apart in the first period to create a 1-1 tie, but Kostitsyn put the Predators in front for good with 8:01 to go in the opening stanza.
That goal came late in the penalty being served by Doug Janik for high sticking.
Kostitsyn gave Nashville a two-goal lead 4:34 into the third while Brad Stuart was in the box for interference.
The three Nashville goals came against Joey MacDonald on 20 shots.