"When you're playing the right way, it is going to be a different guy every night," Minnesota coach Mike Yeo said. "You can't have the same person going out and doing it all the time. But if every guy is doing the right thing then their turn will come."
Prosser netted the winning score for the second time in as many games. The 27-year-old defenseman sent Nino Niederreiter's rebound past Kari Lehtonen's glove for Minnesota's third victory in its last four games.
Asked if it was the biggest goal of his career, Prosser responded: "Yeah, there's not many of them, so it's pretty easy to say."
Koivu is out after he had surgery to repair a broken ankle. Parise has been sidelined by a foot injury. But the Wild (27-19-5) have won seven of nine since a six-game losing streak in December.
Erik Haula and Ryan Suter also scored for the Wild, and Darcy Kuemper stopped 33 shots in his fourth straight start. Kuemper is 4-1 in five starts since Jan. 7 in place of hurting Niklas Backstrom and Josh Harding, who is out due to complications with his multiple sclerosis.
"It's disappointing," Chiasson said. "This was probably one of our best games defensively, as far as scoring chances given to the other team. It's just a little play there, and that's tough. We battled through and it would've been nice to get a win here."
Haula's first career goal put Minnesota up 1-0 at 7:11 in the first. After knocking the puck away from Erik Cole along the boards, the former University of Minnesota standout skated ahead and beat Lehtonen on the stick side.
"It took me a while to kind of get over it and chills to go away, but it was unbelievable," Haula said.
But the Wild had only three shots in the opening period, with Dallas outworking the home team on offense. Minnesota failed to get a shot off during an extended power play late in the period when Shawn Horcoff received a double minor for high-sticking.
But Kuemper's solid play in net offset Dallas' aggressive offensive play.
"We were under siege a lot of the night but there was opportunities where he came out and made a play and relieved some of that pressure," Yeo said of Kuemper. "When you see that kind of confidence, for sure your team feeds off that."
Dallas tied it at 11:57 of the second on Garbutt's 10th goal of the season. Garbutt picked up a rebound off Vernon Fiddler's shot in front of the net and knocked it in past Kuemper.
Yeo then called for a timeout to rally his team, and his players responded. Minnesota reclaimed the lead when Suter wristed a shot from the left circle through a crowd and past Lehtonen at 14:09.
Chiasson's power-play goal for Dallas tied it with just over 10 minutes to play.
The game capped the eighth annual Hockey Day Minnesota that included three high school hockey games on an outdoor rink in the Minneapolis suburb of Elk River, and the University of Minnesota men's team against Ohio State.
Wild D Jonas Brodin left early in the third after taking a stick to the face from Dallas F Antoine Roussel. Play was halted while the arena crew scraped blood from the ice. Brodin returned to the bench in the third. ... Dallas RW Valeri Nichushkin was a healthy scratch for only the second time this season. ... The Wild reassigned D Jonathon Blum to Iowa of the AHL earlier in the day. ... Haula's goal marked a third straight Hockey Day Minnesota game in which a Wild player scored his first career NHL goal. ... The announced attendance of 19,192 was the largest of the season for Minnesota.