Winning pitcher Tony Urban (3-1) allowed only three hits in seven innings and struck out three. Stout pitched the eighth and ninth innings to earn the save and tie Todd Hornsby's school record.
The Bruins had runners on second and third with one out in the ninth, before Stout got a strikeout and a flyout to end the game.
Jacksonville State scored its two runs in the fifth inning. The Gamecocks received three walks, then Griff Gordon delivered a two-out single to left field, driving home Cal Lambert and Michael Bishop.
Jacksonville State (27-21, 18-6 Ohio Valley Conference) is tied for second in the league with Belmont (32-14, 18-6). They trail Tennessee Tech, which is 17-4 in the OVC.
The Gamecocks will host Auburn on Tuesday at 6:30 p.m.