Odum had seven rebounds for the Sycamores (14-9, 5-7), who outscored Drake 36-16 in the paint en route to winning their third straight game.
Both teams were playing their first games since having to work overtime last weekend. Drake needed three extra periods to outlast Wichita State 93-86 on Saturday night, and Indiana State beat Evansville 90-81 in double overtime on Sunday.
Lathan's layup gave the Sycamores a 53-48 lead with 2:09 remaining. The Bulldogs failed to score on their next two possessions.
Indiana State made eight free throws in the last 51 seconds, providing the cushion to withstand late 3-pointers by Rice and Kurt Alexander for Drake.