Keenum entered Thursday's game with 130 and found Charles Sims on a 41-yard pass in the third quarter for his 135th touchdown to set the record and put Houston (No. 17 BCS, No. 18 AP) ahead of Rice 45-20.
He hit Patrick Edwards on a 57-yard pass for his first touchdown at the end of the first quarter. In the second quarter Tyron Carrier caught one for 21 yards and Edwards grabbed the third one for 64 yards. Justin Johnson's 18-yard reception just before halftime tied the record.
Keenum, who became the FBS career leader in total offense last week, moved ahead of former Texas Tech quarterback Graham Harrell for the record for TD passes.
Earlier in the game, Carrier tied the NCAA record for career kickoff returns for touchdowns with his seventh.
Carrier returned the opening kickoff 100 yards for a touchdown on Thursday night against Rice. Carrier dashed down the right sideline and darted around the defense virtually untouched before outrunning the kicker for the score.
He shares the record with C.J. Spiller of Clemson who had seven kickoff returns for touchdowns from 2006-09.
Carrier also returned the opening kickoff 99 yards for a touchdown in a 73-14 win over Rice in 2009.
Information from The Associated Press was used in this report.