With the win, the Cougars moved to 4-1 on the year and 2-0 in American Athletic Conference play.
Postma was 19-of-27 passing for 172 with two touchdowns and had a 22-yard scoring reception.
SMU (4-2, 1-1 American) took a 12-7 lead on a 37-yard field goal by Josh Williams early in the second quarter, but the Cougars answered with a nine-play, 76-yard drive that gave them the lead for good when D'Eriq King hit Postma on a reverse pass to regain the lead.
Postma then found Ellis Jefferson with a 3-yard pass with 1:08 left in the first half to extend the Cougars' first-half lead to 21-12.
SMU opened the third quarter with a 1-yard touchdown run to trim Houston's lead to three, but a career-long 52-yard rush by Catalon put Houston ahead, 28-19. The drive lasted just 31 seconds and totaled 64 yards on two plays.
SMU used a 46-yard field goal to make it 28-22 Houston with 1:46 left in the third, but another rushing touchdown by Catalon - a 16-yarder - sealed the win for the Cougars early in the fourth.
The Houston offensive line held the nation's leading sack team to no sacks and helped the Cougars rush for 265 yards, led by Catalon's career-best game of 177 yards on 22 carries.
Defensively, Houston's Terrell Williams accounted for two interceptions and 31 return yards, one of which came late in the fourth at the 1-yard line when SMU was driving.
Isaiah Johnson notched a career-high 11 tackles to lead all defensive players.