Flames defenseman Blair Jones was hit by a shot in front during the overtime power play and rendered ineffective. With essentially two Calgary defenders on the ice, Doan was set up by Vrbata at the left faceoff circle and ripped a one-timer past goalie Leland Irving.
"We've been scoring by committee lately, and it's important that everyone contributes," Doan said. "Vrbata gave me a real soft pass and I let it go."
Calgary scored the tying goal on Olli Jokinen's tip-in just more than 4 minutes into the third period.
Tyler Pyatt scored less than 8 minutes into the game to stake Phoenix to an early lead. Mike Smith finished with 23 saves.
At the start, the Coyotes benefited from Irving's long layoff. Making his first appearance since Jan. 5 and first start since Dec. 30, the Flames' goalie showed his rust.
With the game scoreless near the halfway mark of the opening period, Irving faced a 2-on-1 Coyotes rush. Leaning back in the crease, he allowed Pyatt's wrist shot to trickle off his pads and into the net.
Pyatt's seventh of the season put Phoenix in front 7:43 into the game.
"Yeah, I was a little rusty," Irving said. "They seem to shoot from everywhere and it took a while for me to settle into the game. I just wanted to give us a chance to win, and we did come away with a point."
Late in the first, Jarome Iginla came close to tying the game. In alone on Smith, he held the puck just long enough for defenseman Keith Yandle to force Iginla wide, and there was no shot on net.
The Flames lost a key part of their offense when Alex Tanguay, third on the team in scoring, left midway through the opening period. His was struck on the cheek with a puck in front of the Phoenix net and left the ice immediately.
That forced Calgary coach Brett Sutter to juggle his lineup.
Mike Cammalleri, acquired in a trade from Montreal on Jan. 12, replaced Tanguay and joined a line with Iginla and Jokinen. Sutter started the second period with Iginla between Jokinen and Tom Kostopoulos. Eventually, Tanguay, wearing a full face mask, rejoined the line with Iginla and Jokinen 5 minutes into the second period.
After the game, Sutter said Tanguay had blurred vision and was kept out during the final minutes of the third period and overtime.
The near air-tight Phoenix defense limited the Flames to four shots through 12 minutes of play and spent the night clearing pucks and bodies from in front of the net. That enabled Smith to make routine saves and keep Calgary off the scoreboard.
Before Jokinen's tying goal, Smith's biggest challenge came late in the second period when he stopped Tanguay in front during a Flames power play.
"There was a point here to be had, and we gave it away," Sutter said. "We knew this game would be tough, and I'm proud the way the guys battled back to tie it."
With five points separating six teams for playoff positioning in the Western Conference, Sutter put the stretch drive in perspective.
"We just have to keep pushing," he said. "It's all right there for whoever wants it."
Pyatt's goal was his third in his last 27 games. ... Smith appeared in his 43rd game of the season, a career high. . The Flames play five of their next seven at home.
Copyright by STATS LLC and The Associated Press
Arena: Gila River Arena Location: Glendale, Arizona
Referees: Brian Pochmara, Brian Pochmara Linesmen: Derek Amell, Brad Lazarowich
Attendance: 10,048 (58.7% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Taylor Pyatt (7)
Assists: Rostislav Klesla, Kyle Chipchura
No scoring this period
Olli Jokinen (17)
Assists: Mark Giordano, Scott Hannan
Shane Doan (16)
(Power Play) Assists: Radim Vrbata, Ray Whitney