LaBarbara notched his fifth career whitewash, stopping 13 Stars shots in an active second period and surviving a pair of Dallas power-play opportunities.
Turris broke the scoreless tie midway through the second period with a wristshot off a rebound from Shane Doan's breakaway attempt.
Phoenix also honored defenseman Ed Jovanovski, who played in his 1,000th NHL game and became the 256th player in NHL history to reach the mark.
Kari Lehtonen absorbed the hard-luck loss for the Stars, stopping 24-of-25 shots.
2014: The Year in Music [PHOTOS]