Egor Koulechov hit three free throws after getting fouled on a 3-point attempt to trim South Alabama's lead to 62-60 with 4:27 to play.
Back-to-back layups by Boyanov and Tafari Whittingham pushed the lead to six when Marcus Evans made a 3 to make it 66-63. Wittingham's free throw made it 67-63 before Marquez Letcher-Ellis made one of two free throws and Connor Cashaw's 3-point play tied it at 67 with 1:03 left.
Whttingham made a layup, but Evans missed a chance to tie it and South Alabama (5-6) converted 3 of 4 free throws in the last 29 seconds to hold on for the win.
Evans had 22 points and seven rebounds for Rice (5-7). Max Guercy had six of the Owls' 10 steals on the night.