Bergeron finished with 110 yards on 15 carries.
Malcolm Brown piled up 105 yards and scored once on 14 rushes for Texas (1-0), which took its 13th straight season opener under Coach Mack Brown.
David Ash found Jaxon Shipley on a 16-yard TD pass among his 20 completions on 27 attempts en route to 156 passing yards.
Shipley made seven grabs in all for 45 yards.
The Cowboys (0-1) were led by Brett Smith, who was 16-of-28 for 276 yards, two touchdowns and two interceptions.
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'
Couple mistakenly served bag of cash at McDonald's drive-thru