Hedberg denied Kyle Calder leading off the eighth round before Morrow's shot went in off Khabibulin's arm.
Stars coach Dave Tippett was running out of forwards when he turned to Morrow, who gets most of his goals from in front of the net.
"It was fun for the fans but not for the coach," Tippett said after his team's longest shootout by four rounds. "We were almost done with the forwards and down to the defensemen, so it's good that Morrow scored."
Morrow said he simply wanted to get the puck on net.
"I did a low blocker shot," Morrow said. "It bounced around his armpit a little bit. I just didn't want to make it too complicated."
Dallas won the first three meetings with Chicago by a combined 21-6, but the Blackhawks gave a much better effort in the finale of the season series.
"It seems every time we've played a big team we've wasted a challenge," Chicago's Martin Lapointe said. "Tonight we played a good team and raised our level of play. Our goal tonight was to keep it simple. Our guys didn't quit."
Tippett wasn't surprised that the Blackhawks were able to extend the Stars.
"We have had some games that have been lopsided against them so human nature tells you that it's in the back of your mind," Tippett said.
Zubov and Philippe Boucher had power-play goals to help the Stars win for the ninth time in their last 11 games. Calder and Mark Bell scored for the Blackhawks, who've lost four straight and seven of eight.
With the score tied at 1, Boucher's shot from a sharp angle hit off the stick of Chicago defenseman Jim Vandermeer and past Khabibulin at 6:40 of the third period for Boucher's 15th goal of the season.
Seconds earlier, Hedberg skated far out of his net and made a sliding poke-check to knock the puck away on a breakaway by Bell.
But Chicago tied it at 2 with less than a second remaining in regulation when Bell notched his 22nd off a goal-mouth feed from Calder.
Stars No. 1 goalie Marty Turco got the night off after posting his NHL-high 37th victory Wednesday night, and Hedberg stopped 24 shots in his 15th game of the season.
The Blackhawks picked up two penalties in the opening 1:08 of the second period after a scoreless opening period and Zubov cashed in from 40 feet at 2:02 with his 11th goal of the season while the Stars skated 5-on-3.
Chicago tied it at 1 when Calder converted a cross-ice pass from Radim Vrbata at 2:10 of the third period for his 18th goal of the season.
Notes: Stars RW Bill Guerin, out with a fractured orbital bone sustained in a game on March 6, will wear a shield to protect his face when he returns. He has started light workouts but there is no timetable for his return. The Stars anticipate he'll be back before the playoffs. ... Dallas is 12-0-1 in its last 13 meetings with Chicago. ... The Stars were 9-1 winners in Chicago's previous trip to Dallas on Nov. 4.