The Cougars' scheduled season opener against Texas-San Antonio was postponed last week because of Hurricane Harvey.
Jjunior quarterback Kyle Allen had his way with the short pass, connecting on 25 of 32 passes for 225 yards and one touchdown. He threw two interceptions.
Linell Bonner was his favorite target, getting nine catches for 59 yards and a touchdown.
Dillon Birden's 3-yard touchdown late in the first half gave the Cougars a 17-10 lead.
The Cougars had just enough, stuffing an Arizona drive with 3:49 left in the fourth quarter when junior safety Garrett Davis intercepted a Khalil Tate pass.
Arizona had a last-ditch possession with less than three minutes left but Arizona failed to get a first down behind quarterback Brandon Dawkins, who started the game, was later replaced by Tate and was brought back for the final drive.
Dawkins rushed for only 6 yards on fourth-and-10, turning the ball over to Houston on Arizona's 23-yard-line. Houston then ran out the clock.
Houston finished with 383 total yards to Arizona's 371.
Dawkins had a rough night, fumbling away two balls, including one that resulted in a safety for Houston in the third quarter that gave the Cougars a 19-10 lead. Later in the quarter, he drove Arizona to the 4-yard line, only to overthrow a wide open Tony Ellerson in the end zone.
Arizona had to settle for a field goal that made it 19-13 with 1:48 left in the third quarter.
Over throws were Dawkins' biggest issue. That and Houston's ability to slow Arizona's offense.
Sophomore Tate replaced Dawkins, managing to lead Arizona to a field goal to make it 19-16 with 8:32 left in the fourth quarter.
Dawkins finished 17 of 29 for 178 yards, while Tate was 5 of 8 for 41 yards and an interception.