Jordan Bachynski added 12 points and five blocked shots for the Sun Devils (20-7, 9-5 Pac-12).
Brock Motum scored 17, Royce Woolridge 16 and DaVonte Lacy 15 for the last-place Cougars (11-16, 2-12) in their seventh loss in a row.
Arizona State finished the first half with a 14-3 run to lead 36-22 at the break and was up by as many as 21 before Washington State cut it to eight in the late going.
The Sun Devils, one of the worst free-throw shooting teams in the country this season, sealed the win with eight straight in the final 1:35.
The game turned in the final 5:49 of the first half. After Felix made one of two free throws to put Arizona State ahead 23-19, Carson brought the crowd to its feet when he drove from midcourt down the lane and soared for a dunk. On the Sun Devils' next possession, Carson found Evan Gordon open for a 3-pointer. Another 3, by Jonathan Gilling, capped a 9-0 spurt and made it 31-19 with 4:18 left in the half.
Motum's three-point play cut the lead to 31-22, but Gilling sank another 3 to boost the lead to 34-22 with 2:34 left, then Carson made both ends of a one-and-one with 7.6 seconds to play to make it a 14-point game at halftime.
Washington State shot 33 percent in the first half (9 for 27) and was 3 of 12 on 3s. Arizona State, on the other hand, shot 52 percent and made 5 of 12 3-pointers.
Felix scored the first five points of the second half to put the Sun Devils ahead 41-22.
Felix's 3 gave Arizona State its biggest lead, 49-28, with 12:15 to go, but the Cougars scored the next eight points to cut it to 49-36 on Lacy's 3-pointer with 9:47 to play. Carson's 3 boosted it to 56-40 with 7 1/2 minutes to go but the Cougars scored the next five, cutting it to 56-45 on another 3 by Lacy with 7:05 to go.
Woolridge's layup made it 61-53 with 2:24 to play, but Carson made both halves of a one-and-one to boost it back to 10. After Woolridge scored again inside, Gordon made two free throws to put Arizona State up 65-55 with 1:18 to play. Motum cut the lead to 65-57, but Felix made four straight free throws to end the game.
Arizona State reached 20 wins for the fourth time in coach Herb Sendek's seven seasons at the school.