Stastny got both scores during a second-period outburst in which the Avs reeled off four unanswered goals to put the game away.
In 26 career games against the Stars, the Avalanche center has amassed 16 goals and 14 assists for 30 points.
Semyon Varlamov stopped 31-of-33 shots.
Tyler Seguin notched a goal and an assist and Kevin Connauton also scored for Dallas, which dropped a third in four games.
Kari Lehtonen made 42 saves for the Stars.
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