The Wild, who have won two of three, sent Nashville to its fourth loss in five games.
Cullen and Wild captain Mikko Koivu both scored in the shootout. Craig Smith was the only Nashville shooter to score in the tiebreaker against Niklas Backstrom.
Zach Parise gave the Wild a 1-0 lead in the second period. David Legwand tied it for Nashville in the third.
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