LaBarbera was making his fifth straight start in place of Coyotes No. 1 goalie Ilya Bryzgalov, who is out because of a lower-body injury.
LaBarbera, who had surrendered 15 goals over his previous four starts, turned aside a penalty shot by Loui Eriksson in the second period.
The Stars pulled goalie Kari Lehtonen for an extra attacker with 1:32 left, but Eriksson missed the net on a close-range attempt.
Phoenix, which has managed one goal in each of its last three games, went 2-3-1 on its trip.
Lehtonen made 25 saves for the Stars, who had been 10-0-3 in their previous 13 home games.
Turris gave the Coyotes all the offense they needed when he converted the rebound of Shane Doan's shot at 10:46 of the second period.
Phoenix nearly made it 2-0 in the second period during a four-minute power play after Dallas' Brian Sutherby's stick struck Wojtek Wolski in the face.
Martin Hanzal of the Coyotes appeared to score on a deflection at 15:11, but the goal was disallowed after a video review. Officials ruled that Hanzal's stick was above the crossbar, and the Stars went on to kill the penalty with several nice stops by Lehtonen.
Eriksson was unable to convert his penalty shot at 6:48 of the second period after Phoenix's Ed Jovanovski interfered with Eriksson on a breakaway.
NOTES: Jovanovski played in his 1,000th NHL game. ... Stars D Nicklas Grossman and C Tom Wandell were scratched due to illness. ... Dallas was shut out for the third time this season. Phoenix had its third shutout. ... The Coyotes had allowed six power-play goals the previous two games.