Odunsi's free throw at the 2:44 mark gave the Huskies (3-2) a 56-49 lead but Seth Gearhart made four free throws and Max Guercy a layup to pull Rice (1-6) within one with 23 seconds left, setting up the frantic closing seconds.
After Odunsi's free throws, Gearhart missed two 3's and Guercy a third as time ran out.
Colter Lasher had 16 points for Houston Baptist, which led 32-25 at the half by closing on an 8-2 run.
Houston Baptist pushed the lead to 10 midway through the second half but Rice scored eight straight.
Gearhart had 17 for Rice, which made 13 of 15 free throws while the Huskies were going 9 of 18.