The victory moved Dallas into the eighth and final playoff position in the Western Conference for the first time this season as the teams headed into the NHL's Olympic break.
The Stars and Coyotes have identical 27-21-10 records for 64 points, but Dallas holds a tiebreaker over Phoenix.
Dallas limited the Coyotes to 27 shots, 13 in the third period. Lehtonen shut them out until Radim Vrbata scored on a power play with 15 seconds remaining.
Whitney scored just 2:10 into the game. Nichushkin scored at 19:26 of the second period, deflecting in a shot by Trevor Daley.