Cal Clutterbuck, Warren Peters, Nick Johnson and Cullen scored in regulation for the Wild, who led 4-2 in the third period before Logan Couture and Patrick Marleau scored with a span of 42 seconds for the Sharks to set up extra time.
Josh Harding gave up four goals on 38 shots for the Wild, who have just two wins in their last 13 games.
Dan Boyle had a goal and two assists, Marleau added an assist and Benn Ferriero scored for the Sharks, who had come into the game with a four-game winning streak.
Antti Niemi had 27 saves, but was unable to come up with the needed stops in the shootout.
Wisconsin business offering 'therapeutic cuddling' forced to close
Exploding whale video goes viral on Internet