Dubnyk stopped another shot by Benn with 1:00 left, after the Stars pulled goalie Antti Niemi for an extra skater.
Minnesota beat Dallas for the first time in four games this season. The Wild had lost the two most recent meetings despite taking leads of 3-0 and 2-0.
Dallas had an apparent goal disallowed in the first period. Video review indicated that Antoine Roussel had interfered with Dubnyk.
The Western Conference-leading Stars have lost four of their past five games. Niemi made 23 saves.
Minnesota improved to 22-11-8 with 52 points -- Dallas has 62 points.
The Stars' John Klingberg sent a shot from the slot past Dubnyk 4:46 into the game, but the Wild successfully challenged Roussel's interference.
Dallas outshot Minnesota 11-6 in the first. During the period's only power play, the Stars' Tyler Seguin hit a post and had another shot gloved by Dubnyk.
Niemi stopped Minnesota's best scoring chances in the first -- short-range shots by Mikael Granlund a minute in and Justin Fontaine with 3 1/2 minutes left.
The Wild became the aggressors in the second period. They outshot Dallas 6-3 in the first 6 minutes, including the game's first goal.
Carter skated in from the left corner to knock in a loose puck in front of the net at 3:34.
Minnesota almost had another score at 9:29. From a scramble in front, the puck went under Niemi. A video review showed that the puck didn't completely cross the goal line.
The Wild did increase the lead to 2-0 at 11:20. The Stars' pass out of their zone hit the stick of Minnesota's Charlie Coyle. The puck went back to Vanek, who skated in on Niemi alone and tucked a backhand into the net for his 13th goal.
In the second, the Wild outshot Dallas 15-8. The Stars had a 16-4 advantage in the third.
NOTES: In 17 career games against Dallas, Vanek has 17 points (8 goals, 9 assists). In his previous 11 games overall this season, he had scored only an empty-net goal. ... The Wild went 0 for 1 on the power play, and Dallas was 1 for 2. Minnesota now is 1 for 21 over the last 11 games, and the Stars are 2 for 18 over their last seven games. ... D Mike Reilly, who played high school and college hockey in Minnesota, made his NHL debut for the Wild. ... The loss ended Dallas' five-game home winning streak. Also ended was Stars LW Patrick Sharp's career-best 11-game point streak.