Norville Carey's dunk with 7:28 left gave Southern Miss (5-12, 0-6 Conference USA), which lost its ninth-consecutive game, a 52-42 lead.
A jumper by Carey gave the Golden Eagles a 54-51 lead but a three-point play by Gearhart tied it with 1:08 to play.
Carey found Matt Bingaya for a layup and a 56-54 lead before Gearhart's layup tied it with 41 seconds remaining.
After Bingaya missed a jumper, Jackson made two free throws and Rice led 58-56 with 5 seconds to go.
Bingaya missed a potential tying jumper.
Chip Armelin led Southern Miss with 18 points and Carey added 15.