Sam Steel scored and Jake Oettinger made 26 saves for the Stars (4-3-1), who snapped a four-game winless streak (0-3-1).
Jackson Blake and Sebastian Aho each scored for the Hurricanes (6-2-0), who lost for only the second time this season. Brandon Bussi made 31 saves.
Blake gave the Hurricanes a 1-0 lead at 4:14 in the first period on a shot from the high slot after he won the puck off a face-off to the left of Jake Oettinger.
Aho made it 2-0 at 19:49 on a snap shot from the top of the right face-off circle on a 2-on-1 breakaway with Seth Jarvis.
Heiskanen cut the lead to 2-1 at 1:04 in the second period when his shot from below the left face-off circle ricocheted in off the skate of Hurricanes forward Nikolaj Ehlers.
Steel tied it 2-2 at 5:41 when he tipped in a Thomas Harley point shot on the power play.
Stars forward Roope Hintz left the game midway through the third period after being hit by Taylor Hall in the corner against the boards.
Heiskanen scored to give Dallas a 3-2 lead at 11:18 in the third period on a 4-on-3 power play when his shot deflected off the body of Hurricanes defenseman Mike Reilly and into the net.