Dwayne Smith scored 14 points and Isaiah Armwood had 12 points and 10 rebounds for the Colonials (4-5).
The Wildcats let a seven-point second-half lead slip away but jumped ahead for good on Nino Williams' putback with 3:18 remaining. Rodriguez was fouled while attempting a 3-pointer with Kansas State ahead 58-57, and he sank all three free throws with 1:46 left.
With Kansas state leading 63-60, Rodriguez missed a one-and-one free throw with 26 seconds left. After Joe McDonald's layup cut the lead to one, Spradling made two free throws, putting Kansas State up three with 13 seconds left.