Cullen opened the scoring with a shorthanded goal 9:23 into the contest, tallied late in the opening period to give Minnesota a 2-1 lead and was credited with an assist in the second period on Andrew Brunette's power-play goal.
It was Cullen's first two-goal game of the season.
Niklas Backstrom stopped 36-of-37 shots, helping the Wild win a third straight game for the first time since mid-November. Colorado has dropped two in a row following a six-game winning streak.
Craig Anderson gave up three goals on 14 shots before being replaced by Peter Budaj, who turned aside all 18 shots he faced.
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