Texas quarterback David Ash, who won the starting job over Case McCoy in training camp, was an efficient 20-of-27 passing for 156 yards and a touchdown to Jaxon Shipley.
Bergeron had 110 yards and scored Texas' first touchdown on a 1-yard run in first quarter. He put the game away with a 17-yard TD in the fourth. Brown ran for 105 yards for Texas (1-0).
Robert Herron had touchdown catches of 82 and 22 yards for Wyoming (0-1). Cowboys quarterback Brett Smith passed for 276 yards but threw two interceptions in the second quarter that set up Texas touchdowns.
With 101,142 in the stands watching to see how Ash would settle into the starter's role after a rocky freshman season in 2011, what they saw was a quarterback comfortable throwing short passes and handing off to his bruising running backs.
Texas also got big plays from its defense with two second-quarter interceptions that flipped momentum, and a fourth-down stop inside the Texas 10 after Ash made his only blunder with a fourth-quarter fumbled snap.
The interceptions by Kenny Vaccaro and Carrington Byndum set up Ash's 16-yard TD strike to Shipley and a 7-yard touchdown run by D.J. Monroe. Those scores allowed Texas to turn a 9-7 deficit into a 24-9 lead by halftime.
Ash had said before the first game he'd be happy to spend the whole night handing the ball off if it meant winning. Bergeron and Brown give Texas a duo that can wear hammer defenses over four quarters.
Ash was dogged by interceptions last season and spent most of the game throwing short passes to the sideline and to his running backs to keep drives alive. He played mostly mistake-free until the fumble after Herron's second touchdown had pulled Wyoming within 31-17.
The Cowboys converted one fourth down but were stopped on another when Smith was tackled a yard short of the marker. Bergeron then burst up the middle for a 53-yard run. His second touchdown came when he broke through the line to his left and rumbled to the end zone.
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