Eastern Illinois (8-6, 2-0 Ohio Valley) scored just 17 points before the break - its lowest scoring half of the season - and trailed by 12 after Jacksonville State's Avery Moore hit a 3-pointer with 7:52 left in regulation.
Eastern Illinois' Josh Piper made a layup with 21 seconds left made it 46-all and, after Jacksonville State (7-9) missed a layup with 1 second to go, forced overtime.
Olivier outscored the Gamecocks, who made just 1 of 9 field goals and 2 of 4 free throws, in overtime.
Moore led Jacksonville State with 14 points. D.J. Felder added 10 points and a career-high 13 rebounds.