Kirill Kaprizov and Steel scored, and Filip Gustavsson made 51 saves in his first career Stanley Cup Playoff start for Minnesota, the No. 3 seed in the Central Division . Gustavsson made 17 saves in the first overtime and nine in the second.
"After the second period, I tried going into the locker room and reset there," Gustavsson said. "And then in the third period] and first overtime, it felt like [the puck] started hitting me. Kept the rebounds close to me, and that's when I started to feel more comfortable in the game."
Game 2 of the best-of-7 series will be here Wednesday.
Jason Robertson and Roope Hintz scored, and Oettinger made 45 saves for Dallas, the No. 2 seed in the Central. Oettinger made six saves in the first overtime and six in the second.
"Just a bounce that went their way," Oettinger said. "We had a couple that didn't go our way; that's hockey. I'm just proud of the effort the guys gave."
[RELATED: [Complete Stars vs. Wild series coverage]
Stars forward Joe Pavelski left the game at 12:02 in the second period after a hit from Wild defenseman Matt Dumba.
"No, no I'm not confident for [Pavelski's availability for] Game 2," Dallas coach Peter DeBoer said. "He's OK. He's walking out of the rink on his own OK."
Dumba received a minor penalty for roughing on the play.
"To be honest, I thought it was a clean hit," Dumba said. "Shoulder on shoulder. I don't even know why I got the roughing, probably because I was just in the box already."
Kaprizov gave Minnesota a 1-0 lead at 19:12 of the first period, scoring on the power play after tipping the puck from the top of the crease.
"I didn't like our start," DeBoer said. "You know, sometimes it's easier to start on the road in these Game 1's in the playoffs. Didn't like our start. I thought once we started to play in the latter part of the second period, third period, and the overtimes, we really put a lot of pressure on them. Both goalies were good. We hit a couple of posts. That's playoff hockey, those could go either way."