"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 also 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 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.
Bryzgalov stopped Nick Leddy's low drive from the left point, and then got his left pad in front of Brandon's Saad rebound attempt. Smith popped in the loose puck from a scrum in front.
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 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."