Devan Dubnyk made 33 saves for Minnesota, including 24 in the second period.
Zach Parise scored into an empty net with one minute remaining for the 3-1 final.
"On the bench, you could feel there was something like, 'OK, we scored. Who'd a thunk we were going to score?'" Minnesota coach Bruce Boudreau said. "Then, all of a sudden, we kept going. We didn't want to try to take the foot off the gas, and I don't think we did the last 10 minutes."
Ben Bishop made 33 saves for the Stars (3-4-0), who have lost three straight games.
"Banking one off our guy, that's kind of the difference in the game," Bishop said.
Suter scored the game-winner in his first game back in Dallas since he broke his right ankle against the Stars on March 31.