Kyle Brodziak scored 6 minutes into the contest and Cullen doubled the advantage 10 minutes later, getting behind the Anaheim defense on an odd-man rush and converting a Cal Clutterbuck pass into a goal.
Jared Spurgeon closed out the Wild's scoring with 29 seconds remaining on a slap shot from the left point as Minnesota moved past Edmonton into first place in the Northwest Division with the win.
Niklas Backstrom turned aside 32-of-34 shots to earn the victory.
Cam Fowler scored twice for Anaheim, including with 49 seconds left to cut the deficit to one goal, but the Ducks could get no closer.
2014: The Year in Fashion [PHOTOS]
LeBron James reveals his new Nike shoes