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.
Kim Kardashian, Kanye West reportedly set wedding date
NBC reportedly holds celebs hostage to Jimmy Fallon's show