The sophomore carried the ball 18 times for the Longhorns (1-0), who reached the BCS National Championship Game last year only to be defeated by Alabama.
Garrett Gilbert succeeded Colt McCoy at quarterback, connecting on 14-of-23 passes for 172 yards. Four of them went to Malcolm Williams, who racked up 77 yards.
Texas put the game away in the second quarter when Rice quarterback Taylor McHargue fumbled while being sacked and Keenan Robinson picked up the loose ball and ran 11 yards into the end zone.
Newton followed with a 2-yard rushing touchdown on Texas' next drive to make it 24-3.
McHargue went 6-of-11 passing for 90 yards, one touchdown, and an interception for the Owls (0-1).
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