Neal's wrist shot at 8:51 broke a 2-2 tie that the Oilers had forged with a pair of goals earlier in the period.
The victory was the fourth for Dallas in its last five games. Edmonton has one win in its last 10.
J.F. Jacques and Ryan Jones scored for the Oilers. Nikolai Khabibulin allowed three goals on just 20 shots.
Dallas led 2-0 after two periods but the Oilers erupted in the third period with 18 shots on goal, two fewer than the Stars' total for the entire game.
Jacques got Edmonton on the board at 5:50 of the third period. Jones tied it with a wrister at 7:38.