Roope Hintz scored, and Joe Pavelski and Tyler Seguin each had a goal in the shootout for the Stars (14-8-2), who have won six straight games and have an 11-game point streak (10-0-1). They are 13-1-1 in their past 15 after a 1-7-1 start.
"It was [a big win]," Stars coach Jim Montgomery said. "It's a division rival, and I don't think we were on top of our game like we have been, and we still found a way. We're finding ways to win even when we're not playing at our best. We're not beating ourselves right now."
[WATCH: All Blackhawks vs. Stars highlights]
Olli Maatta scored, and Robin Lehner made 40 saves for the Blackhawks (9-9-5), who have lost three in a row.
"Obviously, you want the two points, but you've got to see stuff you can build on," Maatta said. "I think today the effort, the compete [level], was there. We did a lot of good things, so I think that's something that's really important."
Video: CHI@DAL: Hintz tips Klingberg's shot into the net
Hintz made it 1-0 at 14:08 of the first period when John Klingberg's point shot deflected off his shin pads.
It was Hintz's first game back after he missed seven with a lower-body injury; Klingberg also returned Saturday after missing six games with a lower-body injury.
Maatta scored on a 2-on-1 off a pass from Alex DeBrincat to tie it 1-1 at 17:24.
"I think [Khudobin] cheated over towards me a little bit," DeBrincat said. "[Maatta] was wide open over there. If [Khudobin] was standing in the middle maybe, it's a little different play. But I felt he was inching towards [me], so I gave it over there and luckily it went in the net. He made a good play and good shot."
Khudobin made a sequence of four saves at 2:27 of the third period to keep the game tied. Upon video review by the Situation Room to see if the puck crossed the goal line, the call on the ice stood.
"To be honest, I was 90 percent sure that it wasn't a goal," Khudobin said. "I know [Blackhawks forward Andrew] Shaw was celebrating big-time, but I wasn't hesitating on that one."
Chicago forward Dylan Strome deflected a Maatta shot into the net at 17:00, but the goal was waved off because of a high stick, a call that was also upheld upon video review.
"On the second one, I was 50-50," Khudobin said about the goal possibly counting. "The referee called it right away, but there was replays. I thought it was a little high, but you never know."
Khudobin stopped Blackhawks forward Patrick Kane on a breakaway at 2:20 of overtime.
"There were some pretty big saves both ways, no question," Blackhawks coach Jeremy Colliton said. "We had a couple that maybe could've went in, but that's going to happen. Sometimes you are going to get the bounces and the calls, and some nights you aren't. I thought we did enough to score, and I thought we defended pretty well and got some saves."
Khudobin also stopped both Blackhawks shooters, Jonathan Toews and Kane, in the shootout.
"To be honest, I was hesitating," Khudobin said. "On the first one, I'm lucky that I got the poke check. On the second one, it hit my glove. I didn't know what to expect. I was thinking maybe five-hole or whatever. [Kane's] got quick hands, by the way. But it hit my glove, hit the post, and went out."
Chicago was 0-for-4 on the power play and is 1-for-17 in its past six games. Dallas was also 0-for-4, including a power play at 2:27 of overtime.
Video: CHI@DAL: Seguin rips puck through Lehner's five-hole
They said it
"They're a really good hockey team. They have a really good roster. They're playing really well. I thought we gave them a fight, and yeah, I thought it was a really good game back and forth. Obviously, there's things we have to clean up, [they got] lucky on a couple of plays. We could have probably got one or two more [goals], but it didn't happen and it was the same for them. The [penalty kill] was great today, and we got one point." -- Blackhawks goalie Robin Lehner
"He was the best player in the game. We're very fortunate to have [Ben Bishop] as our starter and have him as 1B or 1A, or whatever you want to call it. It's two years now that we're very fortunate with the goalies we have [as] that backbone of this team." -- Stars coach Jim Montgomery on goalie Anton Khudobin
Video: CHI@DAL: Khudobin makes great save to rob Kane
Need to know
Kane had an assist to give him 21 points (eight goals, 13 assists) in a 12-game point streak. … Khudobin tied his season-high for saves, set in a 2-1 win at the Colorado Avalanche on Nov. 1. … The Stars are 11-for-11 on the penalty kill in their past three games. … Dallas forward Justin Dowling left the game late in the second period (nausea).
Blackhawks: Host the Dallas Stars on Tuesday (8 p.m. ET; NBCSN, SN, TVAS, NHL.TV)
Stars: Host the Vegas Golden Knights on Monday (8:30 p.m. ET; FS-SW, ATTSN-RM, NHL.TV)
Video: Khudobin leads Stars to shootout win vs. Blackhawks