Carl Soderberg had a goal and an assist, and Christian Fischer and Conor Garland scored for Arizona (18-10-4), which went 3-1-0 on its four-game road trip.
Darcy Kuemper made 26 saves in his third straight win for Arizona, which was coming off a 2-0 loss to the Pittsburgh Penguins on Saturday.
WATCH: [All Coyotes vs. Blackhawks highlights]
Jonathan Toews had a goal and two assists, and Robin Lehner made 44 saves for the Blackhawks (12-12-6), who won their previous two games. It was Lehner's first loss in six games against the Coyotes.
"Step back I thought from the last two games, for sure," Chicago coach Jeremy Colliton said. "Especially the second and third, I thought we had a tough time making plays, exiting the zone, we're not able to establish zone time and momentum, and they were. Robin was excellent to get us the point."
Each team killed a penalty in overtime before Nick Schmaltz and Garland scored in the shootout on Arizona's first two attempts. After allowing a goal to Patrick Kane in the second round, Kuemper made the save against Kirby Dach in the top of the third to end the tiebreaker.
"Our PK's always great," Kuemper said. "We're confident when we're short. You never want to be down 4-on-3 late, but those guys did an incredible job in front of me. We were able to kill it off and hang in there."
Schmaltz, who was traded to the Coyotes by the Blackhawks on Nov. 25, 2018, had two assists in his first game against his former team.
Toews put the Blackhawks up 1-0 53 seconds into the game when he redirected a backhand pass from Calvin de Haan.
Alex DeBrincat scored his third goal in as many games on the power play to make it 2-0 at 8:40.