Glenn had 15 points and 10 rebounds for the Spartans (8-5, 2-0 Atlantic Sun Conference), who hit 23 of 27 free throws for their best percentage of the year (85.2).
Upstate trailed 50-47 with 11:06 to play when Adrian Rodgers hit a 3-pointer and Glenn made a layup, followed by a free throw a second later for a 53-50 lead with 9:59 left.
After the Eagles (3-7, 1-1) trimmed the lead to one, 55-54, the Spartans used a 13-4 run over a span of 3:06 to lead 68-58 with 4:47 to play.
Florida Gulf Coast got to 74-71 with 41 seconds left, then Glenn made a layup and Upstate made 4 of 6 foul shots.
Bernard Thompson's 16 points led the Eagles.