Marian Hossa also scored for Chicago, which improved to 6-1-2 in its last nine games despite squandering a pair of two-goal leads.
Toews, Chicago's captain, has points in five straight games, with three goals and four assists during that span. The timing couldn't be better as the defending Stanley Cup champions try to maneuver back into playoff position in the congested Western Conference standings.
"We've come out with great starts the last couple of games," Toews said. "It's right there in front of us. We've got to carry that effort right through to the third."
The Blackhawks, however, regrouped in the scoreless third before Toews won it in the shootout. He converted Chicago's first attempt against Ilya Bryzgalov, while Blackhawks goalie Corey Crawford stopped Scottie Upshall, Radim Vrbata and Kyle Turris.
"We've had a lot of similar games," Toews said. "You've got to be confident going into the third period. That's what we've told ourselves."
Crawford made 26 saves in regulation, while Bryzgalov stopped 38 shots for the Coyotes.
"We had a great first period, a lot of momentum," Chicago coach Joel Quenneville said. "We lost a little of it at the end of the second, but had a good third period.
"Crow [Crawford] was solid in the shootout. A couple of goals, we screened him ourselves."
The Coyotes are 0-2-1 following an eight-game winning streak, but were happy to come up with a point as they finished a five-game road trip.
"It was huge for us to get at least one point and give ourselves an opportunity to get two in the shootout," Yandle said. "But they've got a pretty high-skill team, and they're tough to beat in the shootout.
"Just for us to get one point out of that game, being down, it's a big moral victory for us."
The Blackhawks agreed to a five-year contract extension with defenseman Brent Seabrook earlier in the day. The deal reportedly will pay $5.8 million annually to the 25-year-old, Chicago's No. 2 defenseman behind Norris Trophy winner Duncan Keith.
Toews scored twice in the first period as Chicago dominated the first 20 minutes. His shot from a sharp angle off the right wing boards slipped through several players and past Bryzgalov just 2:52 in to open the scoring.
Toews deflected in Seabrook's screened shot from the right point during a 5-on-3 Chicago power play with 3:05 left in the period to make it 2-0.
Aucoin cut it to 2-1 at 2:13 of the second on a low shot from the top of the slot following a faceoff in the Chicago zone.
Hossa's power-play score at 8:14 made it 3-1. Bryzgalov made a pad save on Hossa's initial shot from the slot, then the rebound popped out to Patrick Kane at the right side of the crease. Kane lost control of the puck, but Hossa swept in and scored.
Phoenix responded with two power-play goals from faceoffs late in the second to tie it at 3.
Yandle's quick shot through traffic from the top of the slot landed in the lower right corner of the net with 8:49 remaining in the period to cut it to 3-2.
Hanzal tied it with just under 3 minutes left in the second when he batted in a rebound from the crease.
Both teams had several prime chances in the scoreless third.
- Chicago D Jordan Hendry left early in the third after being hit by Shane Doan along the boards. Quenneville said Hendry sustained a lower-body injury.
- Seabrook has played his entire career with the Blackhawks, and was their first-round draft pick (14th overall) in 2003.
- Phoenix LW Scottie Upshall returned after serving a two-game suspension for an illegal hit on Philadelphia's Oskars Bartulis.
- Coyotes LW Ray Whitney returned after missing three games with a lower-body injury.
- Aucoin returned after missing five games with a lower-body injury.