It took Rozier just two seconds to score the opening basket of the game, Louisville bolted to a 12-2 lead and went on to outscore the Cougars in the opening half 57-28.
The defending national champion Cardinals advanced to 15-3 for the season and are 4-1 in the American Athletic Conference. Cincinnati is the only unbeaten team in league play at 5-0.
With the game having turned into a rout, Wayne Blackshear came off the bench to lead Louisville in scoring with 23 points. Smith added 18 points in 25 minutes, Luke Hancock had 10 and Rozier chipped in nine.
Louisville hit 15-of-28 from 3-point range.
Houston made just 16 field goals and turned the ball over 15 times. The Cougars (10-7, 2-2 AAC) were led by TaShawn Thomas with 16 points and 11 rebounds.