Chris Kendrix and Loomis Gerring each hit a pair at the line in the final :23 to push the Missouri State lead to five and the Bears held on.
Austin Ruder had 20 points, including five 3s and Kendrix added 14 points and eight rebounds for the Bears (11-18, 5-12 Missouri Valley).
Bradley scored the first seven points of double overtime, but Missouri State scored seven of the next eight points capped by a tying 3-pointer from Gavin Thurman with 1:16 left.
After three free throws from Sutherland put Bradley up three, Ruder hit a tying 3-pointer with 6 seconds left to force a third overtime.
Josh Cunningham had 15 points and a career-high 21 rebounds, 11 on the offensive end, and Tramique Sutherland added 20 points for the Braves (8-22, 3-14).