Barnes, who beat Niklas Backstrom for the winner, was the fifth player to score out of 18 shots in the extra session. Pascal Dupuis lost the puck when Marty Turco poked it away on the last chance for Minnesota, which lost its fourth straight game at home.
In regulation, Niklas Hagman scored for Dallas. Mikko Koivu answered in the third for the Wild, which lost top goalie Manny Fernandez in the first period to a mild left knee sprain.
Turco made 25 saves before the shootout and became the franchise's career victory leader with 161, passing Ed Belfour.