
From 2-2, Francis Bouillon scored at the end of the second period and Craig Smith at the start of the third to put the Coyotes 4-2 up, but Oliver Ekman-Larsson and Raffi Torres ensured the game went to overtime.
That still could not separate the teams, but Kostitsyn was the only man to succeed in the shoot-out.
Earlier, Martin Erat and Mike Fisher had scored to put Nashville 1-0 and 2-1 up, with Daymond Langkow and Keith Yandle levelling.
Overtime was also needed in Buffalo, where the Sabres beat the Montreal Canadiens 3-2.
Tyler Ennis scored twice for Buffalo before Tyler Myers got the winner in OT.
Erik Cole and David Desharnais were on target for the visitors.
Gabriel Landeskog scored an overtime winner as the Colorado Avalanche beat the Anaheim Ducks 3-2, while the San Jose Sharks beat the Edmonton Oilers by the same scoreline.
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