The Wild, trailing 3-1 in the second period, began their comeback when Charlie Coyle scored with 6:07 left in the middle frame, and Parise pulled Minnesota even with a power-play goal early in the third.
After a scoreless overtime, the game went to shootout with Parise and Mikko Koivu each scoring in first two rounds.
Cullen also scored in regulation while Backstrom finished with 28 saves for the Northeast Division-leading Wild, who claimed an eight win in nine contests.
Quick turned aside 25-of-28 shots.
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