Jagr, who joined the Stars after playing for Philadelphia last season, also notched two assists, including one on Loui Erikkson's game-winner in the third period.
Former Coyote Ray Whitney -- another off-season signing -- provided a goal and an assist for Dallas, which has missed the playoffs for four straight seasons.
Kari Lehtonen made 37 saves in the victory.
Defending Pacific Division champion Phoenix got two goals and an assist from Radim Vrbata.
Lauri Korpikoski also scored and Mike Smith surrendered four goals on 26 shots for the Coyotes.