Carson scored 22 points and Carrick Felix added 21 for Arizona State with those two players combining to make 15-of-23 from the field.
The Sun Devils advanced to meet Baylor in the second round after the Bears crushed Long Beach State 112-66.
Kentucky, the No. 1 seed in the quarter of the draw containing Arizona State and Baylor, was eliminated on Tuesday so the winner of the game between the Sun Devils and Bears will be favored to reach the tournament's semifinals.
Arizona State (21-12) survived a scoring drought of more than 5 minutes that spanned the two halves. Detroit still came no closer than nine points in the second half.
Jason Calliste led Detroit (20-13) with 19 points.