Jailyn Ingram made two free throws to pull FAU to 66-65 with six seconds to play. Kassim Nicholson missed two free throws on UAB's next possession before Benjamin stole the ball from Michael Forrest with three seconds to go.
Will Butler and Kassim Nicholson added 14 points apiece for the Blazers (12-7, 3-3 Conference USA), who shot 50% (28 of 56) from the floor but made just one 3-pointer (1 of 12). Benjamin finished with nine points.
UAB hosts Southern Mississippi on Thursday. FAU plays at Charlotte on Thursday.
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