Dallas won its seventh straight to match a franchise record.
Modano got the game-winner on a power play with 3:12 remaining. With the game tied at 1, the Stars had a 5-on-3 advantage when Modano notched the 523rd goal of his career, most in NHL history by a U.S.-born player.
Modano's 16th of the season came on a one-timer from the high slot off a pass from Jere Lehtinen as the Stars roared back after trailing at the start of the final period on a night when Bryzgalov was almost unbeatable.
"You can't get sucked into getting discouraged too bad," Modano said. "He's one of the better goalies. You can't get too cute. The goals that went in for us were pretty innocent shots on net. He played great. That's part of the reason they're where they're at. What can you do? All you can do is try to get shots."
Dallas has a seven-game winning streak for the fourth time in club history. The last time the Stars won seven in a row was Nov. 22-Dec. 5, 1997.
The Stars improved their Pacific Division lead to six points over San Jose and Anaheim.
Bryzgalov denied the first 40 shots he faced before Ott tied it at 1 at 11:45 of the third period on a shot from the left circle that glanced into the net off the goalie's right shoulder.
"I gave my team a chance to win the game," said Bryzgalov, who finished with 42 saves. "That's my job, that's what I have to do. I played well, it was so close to winning the game for our side. But they score with 3 minutes left, so it's kind of a tough feeling to lose the game in the last few minutes."
Ex-Star Niko Kapanen scored for the Coyotes, but Phoenix lost its fourth in a row.
Bryzgalov was on top of his game in the second period, stopping all 19 shots he faced while the Stars had three power plays, before the Stars pulled their third-period comeback.
"You get frustrated, but somebody has to step up," Stars captain Brenden Morrow said. "You've got to find a way to win."
The pressure was on Turco to match Bryzgalov.
"When you've got a close game like that, you don't want to be the one to give one up," Turco said. "That would have been even more disheartening."
The Stars got off to a sluggish start, failing to register a shot in the first 9 minutes. Meanwhile, the Coyotes capitalized on their first power play opportunity of the night when Kapanen notched his seventh goal from the right circle off a setup by Joel Perrault at 6:59.
But as the game wore on, the Stars dominated play.
"Those first 10 minutes we were kind of sleepy," Ott said. "But we picked up and controlled play the rest of the game. It was a team effort. It's not one guy or one line. It seems like we've been getting the job done with whoever we've needed. We were going up against a hot goalie. The only way to beat him is to keep shooting and hope one gets by."
The Stars remain without D Sergei Zubov (groin, foot) and D Philippe Boucher (shoulder). Boucher has resumed practicing with the team but there is no timetable for the return of either Zubov or Boucher. ...Earlier on Monday, the Stars announced the signing of RW Krys Barch to a two-year deal worth $1.15 million. Barch, the team enforcer, was scheduled to be a free agent after this season. ...Rookie C Martin Hanzal (shoulder) is out for Phoenix.