Kelly threw for 275 yards and rushed for 66 yards to help the Sun Devils (6-2, 4-1 Pac-12) qualify for a bowl and win their first road game of the season after two losses.
Connor Halliday had 300 yards passing for Washington State (4-5, 2-4), which has lost four of five. Halliday, who set an FBS record in his previous game with 89 pass attempts in a loss to No. 2 Oregon, was 29 of 54 with two touchdowns.
Marion Grice ran for 94 yards as ASU scored at least 50 points for the fifth time this season. Kelly completed 22 of 31 passes and was intercepted once.
Arizona State scored three times in the first quarter, thanks in part to great field position, and three more times in the second to build a lead the Cougars could not overcome.
The Sun Devils lead the series 25-13-2, having won nine of the last 10 games.
Both teams were coming off bye weeks.