Stastny tipped Erik Johnson's long wrist shot past Stars goalie Kari Lehtonen 10:35 into the final frame and Colorado hung on to snap an 0-2-1 skid.
Stastny also assisted on a first-period tally by David Jones and Jean-Sebastien Giguere turned aside 21 shots for the Avalanche.
Giguere weathered a hectic final minute as the Stars closed the game on a power play and pulled their goalie for a sixth attacker, but failed to produce the equalizer.
The Avalanche improved to an NHL-best 16-5-2 in one-goal games this season.
Lehtonen made 21 saves for Dallas.