The Bulldogs (5-4) trailed 81-72 with 3:38 left before scoring the game's final 16 points.
Jeremy Atkinson, who finished with 18 points for UNC Asheville, scored six during the spurt. The Spartans (6-3) missed their final eight shots.
The Bulldogs shot 62.7 percent from the field (32 of 51), and held SC Upstate to 38.9 percent shooting in the second half (14 of 36). UNC Asheville outscored the Spartans 43-39 in the final 20 minutes.
It was the 250th career victory for UNC Asheville coach Eddie Biedenbach.