Through overtime, Chicago fired a season-high 52 shots at Smith, the NHL's busiest goalie. He has faced a league-high 607 shots in 18 games so far. Still, the Blackhawks needed a shootout to settle this one.
"We knew he was a good goaltender going in," said Brandon Saad, who scored Chicago's first goal. "We wanted to get shots but we also wanted to get to the net, crash and get rebounds, take his eyes away.
"We did a good job of that and got a couple goals there."
Pirri tied the score 4-all 3 minutes into the third, and 4 seconds after a Chicago power play had elapsed. He connected on a midair deflection of Brent Seabrook's drive from the top of the circle.
After blowing an early 2-0 lead and then falling behind 4-3 after two periods, the Blackhawks kept peppering Smith. Chicago came out buzzing in the third, outshooting Phoenix 17-6.
"We had some good looks at it and had some great chances in the third," Blackhawks coach Joel Quenneville said. "The recipe to try to beat him is the number of pucks and traffic and bodies. He still made some big saves in the third period."
Radim Vrbata had the lone shootout goal for Phoenix, 7-1-1 in its last nine games.
Phoenix defenseman Michael Stone scored two goals for the first time in his career as the Coyotes overcame an early two-goal deficit to earn a point.
"It's a tough one, but we found a way to battle through adversity and get a point," Smith said.
In the shootout, Smith stopped Patrick Kane and Sharp shot wide. Toews and Pirri, who made his first career shootout attempt, connected on wrist shots.
"They're so skilled, if you think they're doing one thing, they'll make you look pretty foolish doing something else," Smith said. "I was fortunate to make a couple of saves, but not enough."
Smith made 48 saves and Chicago's Corey Crawford had 26 in a wide-open, back-and-forth game.
"They came at us awful hard in the third period," Phoenix coach Dave Tippett said. "We were down a defenseman or two there. They threw a lot of pucks at our net, so we'll take the point and move on."
The 52 shots allowed by the Coyotes through overtime were a season high.
Phoenix defenseman Zbynek Michalek left after five shifts in the first period with a lower-body injury.
The Blackhawks jumped out to a 2-0 lead in the opening 6:22, but the Coyotes came back to tie it 2-all after the first period.
Saad opened the scoring 2:37 in when he popped in a rebound with Smith down on the ice and a scrum in the crease.
Sharp made it 2-0 at 6:22 with a power-play goal, on a rising shot fired from the top of the slot and through a screen.
The 23-year-old Stone cut it to 2-1 just 41 seconds later when his shot from the right point deflected off Pirri's stick and past Crawford.
Doan's power-play goal, chipped in from the right side of the net with 5:56 left in the first, tied it at 2. Keith Yandle's shot from the point missed the net, caromed off the back boards and straight to Doan.
Hossa put Chicago back ahead 3-2 at 5:05 of the second when he drove to the net unchecked and scored on a rebound of his own shot from left wing.
Korpikoski and Stone then scored 2:11 apart late in the second to put Phoenix ahead for the first time, 4-3.
From the edge of the crease, Korpikoski wrapped in a rebound of Vrbata's shot with 5:05 left in the period to knot it at 3. Stone's second goal, on a high drive from the top of the left circle, sailed past Crawford on the stick side and put the Coyotes in front.
After Pirri scored to tie it at 4, the Blackhawks took control and outshot the Coyotes 17-6 in the third. Smith stopped Hossa from point-blank range with a minute left in regulation.
Chicago dominated the overtime and outshot Phoenix 3-0.
After the game, the Blackhawks announced they had reacquired forward Kris Versteeg, a member of their 2010 Stanley Cup championship team, from Florida as part of trade with the Panthers. ... The Blackhawks recalled 20-year-old forward Alex Broadhurst from Rockford of the AHL and placed C Michael Handzus on injured reserve retroactive to Oct. 25. Handzus has missed the last eight games with an upper-body injury. Broadhurst, from the Chicago suburb of New Lenox, Ill., didn't play Thursday. ... Chicago D Michal Rozsival was scratched. He was hit in the face with a puck on Sunday. ... Phoenix assigned forward Chris Brown to Portland of the American Hockey League. He had appeared in just four games.