The Sun Devils (7-2, 5-1 Pac-12) were trailing 19-7 in the fourth quarter but mounted a comeback to retain their lead atop the Pac-12's South Division.
Kelly's 14-yard scoring toss to Smith with 2:37 remaining put them over the top. The signal-caller also ran for two TDs while going 19-of-31 passing for 144 yards.
Marion Grice led ASU's ground game, picking up 136 yards on 20 carries.
The Utes (4-5, 1-5), coming off an upset of fifth-ranked Stanford, were victimized by two late interceptions tossed by Travis Wilson. He finished with 121 aerial yards on 6-of-21 efficiency.
Dres Anderson hauled down four passes for 100 yards and a touchdown.