Jordan Loveridge added 15 points and eight rebounds for Utah (5-2).
Joel Wright had 17 points and 11 rebounds to lead Texas State (3-4). Vonn Jones added 16 points, and Matt Staff had 16 points and 10 rebounds.
The Bobcats hit a pair of 3-pointers in the final minute, but foul trouble had allowed the Utes to pull away earlier.
Utah took a 62-55 lead when Wright went to the bench following his fourth foul with seven minutes left in the game. With Wright out, Loveridge had two offensive rebounds that led to a pair of layups by Dubois.
The Utes were 14 for 19 on free throws in the second half in what had been a close game.
Utah's last road win was on February 19, 2011 at New Mexico. The Utes had lost 19 consecutive games away from the Huntsman Center ever since.
The Utes next host Boise State Wednesday night at 6:30 p.m.