Bergeron fell one touchdown short of the school record held by Heisman Trophy-winning running back Ricky Williams. David Ash passed for 274 yards for Texas (5-2, 2-2 Big 12), which overcame another dismal effort by its defense.
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Baylor's Nick Florence passed for two touchdowns and ran for two. The Bears (3-3, 0-3) missed a chance to get a third straight win over Texas for the first time in a rivalry dating to 1901.
Texas led 42-31 at halftime and the game was on pace to beat the Big 12 record total of 133 points set when Baylor lost 70-63 to West Virginia on Sept. 29.
Texas also got just its third Big 12 home win since 2010.
Since Texas and Baylor have two of the worst defenses in the country, the offenses had no trouble moving up and down the field while carving them up. The teams combined for 1,132 yards, 56 first downs and 14 touchdowns, and the Longhorns gave up more than 600 yards for the second consecutive week. The bright spot for Texas was holding Baylor to field goals instead of touchdowns on three scoring drives in the second and third quarters which made a huge difference in the end.
The game wasn't settled until Texas recovered a pooch kickoff with 1:57 left after Florence scored his second touchdown to get Baylor within six. But Bergeron converted a critical third-and-1 that helped Texas run out the clock.
The game was played at a blistering pace in the first half as both teams marched at will against defenses desperate just to make tackles.
Perhaps not wanting to crush his defense's confidence from the start, Texas coach Mack Brown chose to start on offense after winning the opening coin toss and freshman running back Daje Johnson scored on an 84-yard run on the first play from scrimmage that set the tone for the night. Like everything else the offenses did, he made it look simple by bouncing a run outside and gliding untouched to the end zone.
The fireworks were just getting started. Bergeron scored on runs of 15, 2, 9 and 4 yards and Texas scored 28 points in the second quarter. Florence passed for two scores and ran for another before Texas took a 42-31 lead into halftime.
Florence connected with Terrance Williams for an 80-yard touchdown pass and his 7-yard toss to Lanear Sampson gave Baylor a 28-21 lead. But Florence floated a pass that was intercepted by Texas' Josh Turner in the second quarter. The turnover set up a 25-yard touchdown run by Longhorns freshman Johnathan Gray.
Texas scored again after a Baylor punt to take a 42-28 lead before Baylor drove to Aaron Jones' 29-yard field goal on the last play of the half to make it 42-31.
On the opening drive of the third, Baylor marched inside the Texas 10. Florence tried to run and fumbled, but the ball rolled into the end zone for Baylor tight end Jordan Najvar to recover for a touchdown.
Baylor forced a punt on Texas' first possession of the third, but had to settle for another field goal by Jones that cut the Texas lead to 42-40 instead of pushing for a touchdown. Texas struck quickly with Ash throwing 67 yards to Mike Davis to set up Bergeron's fifth touchdown, this time from 8 yards out, to make it 49-40.
The Texas defense forced another Baylor turnover at the end of the third when Glasco Martin fumbled at midfield. Texas capped the short drive with Ash's first touchdown pass of the game, a 15-yard strike to Davis, and Texas led 56-43 with 12 minutes to play.