After Peter Mueller scored in the shootout's second round, Mike Ribiero scored on a wrist shot to extend the game. Dallas' James Neal and Steven Reinprecht each scored in the fourth round and Mike Modano's wrist shot was stopped by Bryzgalov before Turris swept in from the left, faked a forehand shot and scored with the backhand to end the game.
"I think they know I'm going to shoot so I tried to change my angle up a little bit," Turris said. "I've practiced it. I just kinda thought about it when the other guys were shooting and thought I'd try it out."
Ilya Bryzgalov made 38 saves in regulation and overtime, then stopped three of five in the shootout for the Coyotes, who won for the first time in six overtime games.
The shutout was Bryzgalov's second this season and seventh of his career.
"It's the NHL. Nothing's easy here," Bryzgalov said. "I just react with the play and follow the players."
Peter Mueller and Steven Reinprecht also scored in the shootout for Phoenix, which has won two straight overall to pull even with Anaheim for sixth place in the Western Conference.
"From a pure hockey point of view it's just a hard pill to swallow to lose a game on contests, so to speak," said Phoenix coach Wayne Gretzky. "It is what it is. Everybody has to deal with it. Probably the people who don't enjoy it the most are the coaches."
Marty Turco made 28 saves for the Stars, who have lost three of their last four.
"It certainly felt like we deserved better," Turco said. "It sucks to get to overtime with these guys when you had a chance to win in regulation."
The Coyotes were inches away from scoring late in the third period. Mueller tried to jam the puck past Turco's outstretched glove with 6:43 left, but Stephane Robidas cleared the puck off the goal line. No goal was called and a video review confirmed the referee's decision.
Turco followed with a glove save of a Turris rebound while sprawled on his right side with 3:14 remaining to preserve the tie.
"When you get a shutout, you think that you should get the win," Stars coach Dave Tippett said. "We didn't. We're in the winning business. We're not in the just-about business."
Notes: Dallas fell to 5-5-1 in overtime this season. ... The last two meetings have gone to overtime. ... Coyotes right wing Viktor Tikhonov was scratched with a hand injury sustained during Phoenix's 3-2 victory Thursday night over Tampa Bay. ... Stars right wing Landon Wilson missed his third straight game with an upper-body injury.