This turned into quite the big victory for the defending Stanley Cup champions.
Chicago goaltender Corey Crawford made 25 saves through overtime and then stopped all three Minnesota attempts in the shootout. Hossa connected on the Blackhawks' second try in the tiebreaker by powering a shot through Ilya Bryzgalov.
"We played the way we had to play," Blackhawks coach Joel Quenneville said. "Playing it simple, advancing the puck, a check-first mentality.
"The way we played our last three games, not coming up with anything, it was important we did things the right way," he added. "Getting two points was what we're all about. We needed that."
It was Chicago's first game since Quenneville said Toews would miss the rest of the regular season due to an upper-body injury. The Blackhawks also were without Kane, another star forward who is expected to miss the rest of the regular season because of a lower-body injury.
"It is nice we can do it without them, but they're a big part of our team for how many years?" forward Bryan Bickell said. "We're just hoping that when they come back, they're healthy and give us a spark. We just need to do what we did tonight through the rest of the games."
The Blackhawks did get Bickell back from an injury, and he put Chicago ahead 2-1 early in the third period. But Minnesota's Erik Haula tied it at 2 with 1:54 left in the third.
Hossa's shot in the tiebreaker lifted the Blackhawks to their first shootout win since Jan. 19 against Boston. They also improved to 6-8 in shootouts.
"Fun to get two points in an overtime game," Quenneville said. "It's something we hadn't seen in a long time."
Ben Smith also scored for the Blackhawks.
Charlie Coyle scored for Minnesota, which moved five points ahead of Dallas and Phoenix in the battle for two Western Conference wild-card slots. Bryzgalov stopped 24 shots through overtime.
"That's a huge point for us," Minnesota coach Mike Yeo said. "Obviously we would have liked to have gotten two, but when you can get a point and you're not completely on top of things, that's a good thing."
Haula scored his fourth goal on a screened shot. He snapped off a shot from the right circle that sailed between the legs of Blackhawks defenseman Niklas Hjalmarsson and then deflected in off Crawford's glove.
Coyle scored the only goal of the first on a quirky tip-in.
Jared Spurgeon drifted in from the right point and fired. His shot hit Hjalmarsson's stick, then deflected off the blade of Coyle's stick and flipped over Crawford's glove at 14:19.
Smith tied it at 2:29 of the second with his 11th goal.
Zach Parise, who leads Minnesota with 28 goals, was hit hard from behind in open ice by Chicago's Brandon Bollig at 8:34 of the second period. After lying face-down on the ice for a minute, he was helped to the dressing room by one of the Wild trainers, but he returned later in the period.
"You could see the shift before (Bollig) was targeting guys with a couple of late hits," Yeo said. "I didn't like it. It was from behind."
Bickell, who was sidelined for six games with an upper-body injury, beat Bryzgalov on a rebound of his initial point-blank attempt at 2:45 of the third period.
Toews got hurt when he was drilled by Pittsburgh defenseman Brooks Orpik in the second period Sunday. Kane was injured March 19 against St. Louis and has missed seven straight games.
Quenneville has said both stars will be ready for the playoffs.
Blackhawks D Johnny Oduya was a last-minute scratch with a lower-body injury. ... Chicago C Joakim Nordstrom, recalled from Rockford of the AHL on Wednesday, played. ... Minnesota F Jake Dowell and G John Curry, both recalled from Iowa of the AHL on Tuesday, were in the lineup. Curry backed up Bryzgalov.