Mario Blessing added 12 points for the Spartans (12-4), who were outrebounded 33-30, but forced 24 turnovers - 12 from steals.
The lead traded hands eight times until Blessing's layup at 2:53 put the Spartans up 24-23 and they led 29-27 at halftime. UMKC outshot the Spartans 45 percent to 37 percent in the half but had 12 turnovers to 5 for SC Upstate.
Martez Harrison hit a 3 and UMKC pulled ahead 31-29 early in the second half, but the Spartans used a 9-2 run to pull away and another 9-2 run to lead 59-47 after two free throws by Fred Miller with 1:52 left.
Frank Williams Jr. led UMKC (5-12) with 15 points and Harrison added 11.