Jonathan Stark hit a pair of free throws to tie the game at 85-85 with 1:06 remaining, but neither team could score until the final second.
Murray State's Bryce Jones hit a 3-pointer at the buzzer to tie the game at 77-77 and force overtime.
Rodriguez scored 21 points and grabbed seven rebounds to lead Southern Illinois (4-3), which shot 50.8 percent from the field on 31 of 61 shooting, including 6 of 24 from beyond the arc. Leo Vincent added 17 points, Armon Fletcher 16 and O'Brien 15 for the Salukis.
Stark finished with 29 points on 9 of 21 shooting from the field and 10 of 11 from the line for Murray State (3-4).
