Ryan Suter and Jared Spurgeon also scored, and Mikko Koivu had two assists for Minnesota, which has won five of its past six games and again recovered quickly from a loss. The Wild (38-13-6) are now 14-3-2 following a loss this season, including eight straight wins. They've lost back-to-back games in regulation just once this season.
Minnesota was 1-for-2 on the power play on Tuesday after going 0-for-5 in a 1-0 loss against Anaheim on Tuesday.
Cody Eakin scored his second goal of the season and Kari Lehtonen made 23 saves for Dallas, which has lost seven of its past eight games and slipped to the third-worst points total in the NHL.
The Stars outshot the Wild 35-26, but Kuemper was strong in net in giving No. 1 starter Devan Dubnyk a night off. Dubnyk leads the league in wins, goals-against average and save percentage this season, leading to Thursday being just Kuemper's 13th game of the season.
Kuemper wasn't beaten until Eakin scored 5:19 into the third period.
Antoine Roussel wristed the puck toward the net, and the puck deflected off Wild defenseman Christian Folin's skate right to Eakin. Minnesota challenged goaltender interference on the play as Patrick Eaves tried to slide between Folin and Kuemper and made contact with the goaltender as Eakin was shooting.
A video review upheld the call on the ice of a goal to snap Kuemper's shutout bid.
Suter started the scoring with his first goal in 12 games with just 0.8 seconds left in the first. Dallas' Curtis McKenzie was called for goaltender interference with 40 seconds left in the period and Suter cashed in on the power play in the final second to tie his career high with his eighth goal of the season.
It's the fourth time in Suter's career he's reached eight goals and the third time in five seasons with the Wild. The tally was also his career-best fourth power-play goal of the season.
Haula scored his 12th of the season 4:51 into the second. Haula was standing in front of Lehtonen as Jordan Schroeder went behind the goal and sent a centering pass to Haula, who quickly snapped the puck in the net.
Spurgeon finished the scoring 13:04 into the third with his eighth of the season.
NOTES: Minnesota D Matt Dumba missed his third straight game with a lower-body injury. Coach Bruce Boudreau said it's likely Dumba won't return until at least Feb. 27 because he hasn't skated and the Wild have a five-day break next week. ... The Wild also scratched rookie F Zack Mitchell. ... Dallas scratched D Stephen Johns (illness) along with F Jiri Hudler and F Lauri Korpikoski. ... Minnesota D Jonas Brodin has been skating on his own and Boudreau said Brodin, on injured reserve with a fractured finger, started taking slap shots. Boudreau still didn't know when Brodin might return. ... The Stars return home on Saturday to face the Tampa Bay Lightning. The Wild's franchise-record, eight-game homestand continues Saturday against Nashville.