Kostitsyn, the Preds' first shooter in the tiebreaker, came in on Coyotes goalie Mike Smith with a series of fakes before sliding the puck between his legs for a score.
Nashville's Pekka Rinne made stops on Ray Whitney, and Radim Vrbata and Mikkel Boedker hit the post on the Coyotes' final shootout try, ensuring a victory for playoff-bound Nashville as it embarked on a stretch of four road games over seven days.
Rinne turned aside 31-of-34 shots to earn the victory.
Smith finished with 31 saves for Phoenix.
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