Smith's shutout was the 20th of his career and was his first victory over the season after missing five games with a pulled groin.
Mikkel Boedker and Nick Johnson scored for Phoenix, which was coming off a 4-3 overtime loss to the Stars Friday.
Richard Bachman turned aside 32-of-34 shots for Dallas, which fell for the fifth time in six games overall.
Reindeer recovered after escaping from Santa during lighting ceremony
Wisconsin business offering 'therapeutic cuddling' forced to close