Dan Fritsche also scored for Minnesota, which was outplayed in the first half of the game, but rebounded to end Chicago's four-game winning streak. The Wild's NHL-leading penalty kill snuffed all eight Chicago advantages.
Harding hasn't played poorly this season. He entered the game with a 1-7-1 record, despite a 2.25 goals-against average and .927 save percentage.
"I think I've played well enough most games, but I wanted to show everybody I could play well enough to win," Harding said. "It's getting that win that's important."
Harding, 3-1 lifetime against Chicago, got the call from Minnesota coach Jacques Lemaire after Backstrom made 35 saves in a 5-2 home win over Detroit on Saturday night.
"He was good," Lemaire said. "We had a lot of scoring chances against us. Some of the saves he made, I don't know how he made them, but he made them. That's the bottom line."
Toews gave Chicago a 1-0 lead midway through the second period, scoring on the Blackhawks' 26th shot. Chicago could have blown the game open by then had it not been for Harding.
"After he just kept making great save after great save, we just wanted to get the two points," Fritsche said. "Not just for us, but for that guy."
Chicago's Cristobal Huet started his sixth straight game and made 22 saves. Nikolai Khababulin, who normally alternates with Huet, is on injured reserve because of a lower-body injury.
Minnesota's Owen Nolan sustained a lower-body injury early in the second period and did not return. "
"It doesn't look good," Lemaire said. "We might have to call someone up."
The Blackhawks returned home following a 3-0 road trip. Entering Sunday's game, Chicago had played 11 of its last 12 games on the road, going 8-3-0 in the road contests and 9-3-0 overall.
The Blackhawks play 16 of their final 24 games at the United Center.
"We outshot them and had more chances," Chicago defenseman Duncan Keith said. " We didn't get outplayed. He (Harding) made some nice saves,
"It was one of those nights the puck didn't go in for us," Chicago coach Joel Quenneville said. "We had some great looks, but Harding had a special night."
Chicago outshot Minnesota 18-5 in the first period and dominated play, but Harding made several tough saves to keep the game scoreless.
Additionally, Chicago's Andrew Ladd hit the post on a point-blank chance midway through the first, Toews misfired on a one-time attempt from the edge of the crease in the final seconds.
Harding was especially sharp early in the second to help the Wild whether 4:18 of consecutive Chicago power-play time, including a 5-on-3 advantage that lasted 1:42. He robbed Chicago's Troy Brouwer with a glove save 2:29 into the second during the 5-on-3.
Toews finally beat Harding at even strength to give Chicago a 1-0 lead at 10:03 of the second.
Harding slid to his right to cover a possible shot from Martin Havlat. Havlat, however, passed to Toews, who fired into an open net from the edge of the crease.
Fritsche tied it 2:45 later with a power-play goal on a deflection in the air that withstood a video review.
Parked in front of the crease, Fritsche redirected Pierre-Marc Bouchard's shot from the top of the slot. The puck struck Fritsche's stick, ruled to be at crossbar height, and caromed downward past Huet.
"I wasn't sure if it would be allowed," Fritsche said. "It was on the borderline. It was a big break for us."
Olvecky gave Minnesota a 2-1 lead at 3:30 of the third. Olvecky, playing in his 11th NHL game, fooled Huet with a low shot from the left circle.
"I just closed my eyes and shot," Olvecky said.
Harding made a sprawling pad save on Dustin Byfuglien in the final seconds to preserve the win.
- Minnesota LW Andrew Brunette missed his second straight game because of a knee injury. Brunette's consecutive game streak ended at 509 on Saturday when he didn't play against Detroit. The streak was the longest among active NHL players.
- Toews has five goals and four assists in his last five games.
- Huet tied a season high for consecutive starts by a Blackhawks goalie.
- Although he's on IR, Khabibulin is listed as day-to-day with what likely is a recurring groin problem. Chicago goal leader Patrick Sharp missed his fourth game because of a left knee injury.