Elias Lindholm appeared to have scored for the Flames at 5:19, but the goal was overturned by a video review after the Blackhawks challenged for goaltender interference.
Kadri did tie the game 1-1 at 13:12 by tapping the puck into an open net.
Andrew Mangiapane put the Flames in front 2-1 with a power-play goal at 3:26 of the second period. He scored from the right face-off circle with a shot that deflected in off the right skate of Blackhawks defenseman Connor Murphy.
Philipp Kurashev tied it 2-2 at 12:09, and Nikita Zaitsev gave Chicago a 3-2 lead at 14:49 with a slap shot from the top of the right circle that deflected in off the right skate off Nick DeSimone.
Blackwell scored a power-play goal at 14:37 of the third to extend it to 4-2. The forward, who made his season debut on Dec. 19 after recovering from sports hernia surgery on March 23, had the second three-point game of his NHL career (three assists for the New York Rangers in a 4-0 win against the Devils on April 15, 2022).
"He's been unbelievable since he's been back," Chicago coach Luke Richardson said. "He was frustrated that it took so long to get back, but he did all the right things to get in shape. I was on the ice one day with the injured guys, and he really pushes, and he does all the right things in the gym and prepares himself, so I'm glad to see him have some success here."
Kadri brought the Flames to within 4-3 at 15:19, scoring from the goal line on the right side.
"I just kind of noticed that [Mrazek] was swimming a little bit," Kadri said. "I don't think he was quite sure where the puck was, so I figured I'd just put one short, low short side and was able to find a hole."
Flames coach Ryan Huska said the result came down to one thing.
"Tonight, we got a lot of individualistic play, and I think that hurt us, especially in the second period," he said. "They were hungrier and harder than we were."
NOTES: The Blackhawks played with 11 forwards and seven defensemen. Forward Rem Pitlick, who was acquired in a trade with the Pittsburgh Penguins on Saturday, could not make it to Chicago in time for the game because of weather conditions. … Forward Zach Sanford, who was claimed off waivers from the Arizona Coyotes on Saturday, had an assist on Blackwell's second goal.