Thomas scored six of Houston's final nine points in the last 2:12 to lead the Cougars.
Brandon Morris added 10 points, Danrad Knowles grabbed nine rebounds, and the Cougars shot 44 percent from the field but only 50 percent from the free-throw line.
Rice hit 10 of 26 from behind the arc but just 4 of 13 from 3-point range in the second half. The Owls shot 30.5 percent from the field.
Houston regained the lead on a three-point play by Thomas with 2:12 to go and upped the lead to 49-46 on another Thomas free throw with 1:50 left.
Ramljak missed a 3-pointer with 1:25 to go, and Jherrod Stiggers hit a jumper on the other end with 57 seconds left to open up a five-point lead.
Ramljak nailed a 3-pointer with 38 seconds remaining to cut the lead to 51-49, and Thomas responded with a layup to increase Houston's lead to four with 26 seconds left.
Keith Washington drove for a layup on Rice's ensuing possession, but Thomas blocked his shot with 16 seconds left. Morris stole the inbounds pass and was fouled but missed both free throws with 14 seconds left.
Max Guercy missed a 3-pointer with 8 seconds left, and Jaaron Simmons made one of two free throws to seal the win.
Trailing 44-38 with 9 minutes left, the Owls used an 8-1 run to grab a 46-45 lead on a Sean Obi jumper in the paint with 2:56 remaining.
The Cougars raced to a 37-28 lead with 15:50 remaining on a Thomas jumper, capping an 8-1 spurt in which Thomas had four points.
Rice jumped to a 17-6 lead in the first 7 1/2 minutes of the game, capped by an Ramljak 3-pointer with 12:34 left in the first half. Ramljak scored nine straight points in the stretch, connecting on three 3-pointers.
Houston responded with a 15-3 run to take a 21-20 lead with 5:45 remaining in the opening half on a LeRon Barnes free throw. Barnes had five points in the run.
The Cougars would take a 29-27 lead into halftime following a circus reverse layup by Jimmie Jones with 36 seconds remaining.
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