Jonas Brodin scored on a power play with 5:39 remaining to tie the game for the Wild, who raised their home record to 12-3-2 and handed the Blackhawks their first loss in their past seven road games.
With Charlie Coyle partially screening Corey Crawford at the edge of the crease, Brodin wound up and sent the puck ricocheting off Blackhawks defenceman Johnny Oduya and past the goalie's outstretched glove.
Crawford made 19 saves.
After nearly taking a 2-0 lead into the second intermission, the Wild suddenly trailed 3-2 after a forceful Blackhawks comeback.
Harding stopped 16 shots and improved to 7-3 in his career against the Blackhawks, who entered the night with an NHL-high 44 points. He's 12-1 at home this season.
The Wild went six consecutive games without a first-period goal until Matt Cooke snapped a shot past Crawford. Zach Parise's score on a power play, the first by the Wild in seven games, in the second period gave the Wild a 2-0 lead they had until 16.6 seconds were left before the end of the middle period.
Joakim Nordstrom deflected the puck behind him through the slot, where Jeremy Morin arrived to reach forward and guide the puck in one motion past Harding, who was coming off a 2-0 shutout of Philadelphia on Monday night. Morin and Nordstrom were both recalled on Wednesday from Rockford of the AHL.
The goal brought a loud roar from the Blackhawks fans in the building and some energy the players clearly didn't leave in the locker room during the second intermission. Just 43 seconds into the third period, Saad checked Ryan Suter off the puck as the Wild's star defenceman tried to carry the puck out of his zone.
Suter tripped, Saad took control with an unabated path to the net and quickly snapped a shot into the top right corner of the net to tie the game.
Less than 6 minutes later, Nino Niederreiter was called for holding. With the Blackhawks holding a one-man advantage, Hossa got loose down low and sneaked a wrist shot over Harding's shoulder for the lead.
After their six-game winning streak on the road, the Blackhawks returned home on Tuesday and lost to Dallas 4-3.
NOTES: Morin and Nordstrom were brought up for depth at forward. Andrew Shaw missed his first game due to an upper-body injury suffered on Tuesday night, and Michael Handzus missed his second straight game because of an illness. ... The Wild improved to 10-1-5 in one-goal games this season to match Chicago's win total in those situations. Only Los Angeles, with 11, has more. .. Mikko Koivu, with 566 career games, passed Pierre-Marc Bouchard for second place on the Wild's career list. Nick Schultz is first with 743.