JoJo McGlaston added 10 for the Aggies (3-0), including two of USU's five free throws in the final 50 seconds to secure the win.
Jared Brownridge scored 11 to pace the Broncos (2-1) but missed a 3-point attempt with Santa Clara trailing 57-54 with 15 seconds to play. Matt Hubbard added 10. Brandon Clark came into the contest averaging 26 points but was held to nine on 3-of-15 shooting.
USU took its first lead in the game at 51-49 with 3:50 to play when Smith hit a 3-point shot. McGlaston had another 3-pointer to keep USU ahead with a 55-52 advantage.
USU was 7 of 17 from 3-point range, while Santa Clara was 6 of 24.