Colorado's Wojtek Wolski converted in the first round, but Turco was perfect the next three rounds to end the Avalanche's unbeaten streak in shootouts at eight -- the third-longest streak in the league since the tiebreaker system was implemented in 2005.
Seconds after Modano's goal, Turco came up with night's biggest stop on Smyth to end it.
The goaltender had given up the tying goal to Marek Svatos with 55.5 seconds left in regulation, spoiling Turco's bid for his 34th career shutout.
"It's an awesome sign," Turco said. "The late goal didn't rattle this crew."
Budaj wasn't pleased with the way he played Ribeiro's attempt.
"It's tough," said Budaj, who finished with 23 saves. "The goal that upsets me was Ribeiro's. I backed up far and didn't adjust to his speed."
In the next round, Modano took his time skating in on Budaj, planning to shoot high.
"I've come too quick sometimes, so I figured to take it slow and wait for the top (of the net) to open up and try to get it above his glove," Modano said.
Turco denied Colorado's Tyler Arnason from close range with a little over 5 minutes left in regulation, and made a sliding save on Smyth's rebound attempt with 11:34 to play. But Svatos' late goal denied Dallas' first shutout of the season.
Once the Avalanche got into the shootout, they figured to have an advantage, having won their first four tiebreakers this season.
Colorado has more skilled players than Dallas, but Turco was able to negate that edge in one of his best performances of the season. Turco began the night with a 3.51 goals-against average, 1.28 higher than his career average.
"I like them better when we win," Colorado coach Tony Granato said of his team's shootout run. "We had good skaters out there. Turco made one big save and made another shot go wide."
The Stars are 4-2 in their last six games, their best stretch of what has been a disappointing start for a team that came within two games of reaching the Stanley Cup final last season.
"They're all not going to be masterpieces right now," Stars coach Dave Tippett said. "It's all about getting points."
Turco had 26 saves and Toby Petersen scored his third goal in four games for the Stars.
Petersen's deflection of Modano's centering pass provided Dallas with the game's first goal at 7:57 of the second period.
Petersen notched his first goal for the Stars and first in two seasons in a 4-3 victory over Edmonton on Nov. 30, and has remained hot with three goals and an assist in the four-game span.
The Stars were without forward Sean Avery, serving a six-game suspension announced by the league earlier Friday for a crude comment regarding his ex-girlfriends dating other hockey players.
Stastny's shot wound up over the goal line with 1:31 left in regulation, but the goal was immediately disallowed because the referees lost sight of the puck. ... Dallas added C Tom Wandell, who was playing for Timra in the Swedish Elite League. The Stars' fifth-round draft pick in 2005 will practice with the team over the weekend and could be on the ice for Wednesday night's game against Phoenix. ... Colorado called up F Chris Stewart from AHL Lake Erie. Stewart was on the active roster. ... Injured and out for the Stars were LW Jere Lehtinen (upper body), LW Brenden Morrow (knee), C Joel Lundqvist (shoulder), C Steve Ott (hand) and D Sergei Zubov (lower body). ... Sitting out for the Avalanche were C Joe Sakic (back), D Adam Foote (knee), C Darcy Tucker (knee), RW Brian Willsie (knee) and RW Ben Guite (ribs). ... Colorado closed out a stretch of nine road contests in a 12-game span, going 6-5-1 during that span. ... The Stars were playing for the third time in four nights.