UMKC (4-2) trailed most of the first half but grabbed the lead in the second and held on to give the Jackrabbits (6-1) their first loss this season.
UMKC was trailing 32-31 at the break but Harrison nailed consecutive treys to start the second half to take a 37-32 lead and the Kangaroos never trailed again.
South Dakota State's Jake Bittle, who finished with 10 points, made four-straight free throws and Deondre Parks sliced in for a layup to tie at 57 with 1:02 remaining. Harrison made seven free throws down the stretch to seal the win.
George Marshall led the Jackrabbits with 14 points. Bittle had seven rebounds and five assists.