HOUSTON (AP) Texas Tech showed improvement in Kliff Kingsbury's third season as coach.
LSU showed the Red Raiders they still have a ways to go.
Texas Tech (7-6) cut the lead to one early in the third before the Tigers (9-3) scored 21 straight points, with two touchdowns by Fournette, to make it 42-20 entering the fourth.
Fournette had two rushing touchdowns of 2 yards, ran for TDs of 43 and 4 yards and took a screen pass 44 yards for another score.
Patrick Mahomes threw for 370 yards and four touchdowns for Texas Tech, but was under heavy pressure all night. He was sacked six times and scrambled away from probably 10 more.
''I knew it would be a challenge, particularly when we had to go into throwing mode every play,'' Kingsbury said. ''It's a good front. It's a good defense.''
Jakeem Grant set a Texas Tech bowl record with three touchdown receptions.
''We got better,'' Kingsbury said. ''I think offensively that type of output is what I expect. I think we will continue to get better as Pat (Mahomes) develops and some of these young receivers get better. We have to continue to improve, and turnover margin still wasn't great.''
It was the 10th 100-yard rushing game and fourth of 200 this season for Fournette, who extended his LSU single-season rushing record to 1,953 yards.
LSU outrushed Texas Tech 384-29 as the Tigers took advantage of a run defense that was one of the worst in the nation.
''Defensively, we still have a ways to go,'' Kingsbury said. ''(Defensive coordinator David) Gibbs has done a good job, but we have to get some guys in here coaching-wise that will fit and then improve talent on that side of the ball.''
Brandon Harris threw for 254 yards with a touchdown and an interception and ran for another score for LSU.
Fournette was stopped for no gain on one carry on LSU's next drive and managed just 2 yards on the next one. But Texas Tech couldn't corral him on his next run and he broke free for 43 yards to give him the LSU record for rushing touchdowns in a season with 20 and push the lead to 28-20.
The Red Raiders were driving on their next possession when Rickey Jefferson intercepted a pass that was deflected on the LSU 1-yard line.
Fournette pushed LSU's lead to 35-20 on a 4-yard touchdown run after the interception. Trey Quinn had a 46-yard reception to set up that score.
''I think that the fact that they scored and made it 35-20 instead of 28-24 or 28, that was a big turning point,'' Kingsbury said.
Harris made it 42-20 when he ran 26 yards for a touchdown with 9 seconds left in the third quarter.
Grant had a 4-yard reception for Texas Tech that made it 42-27 early in the fourth.
Fournette sailed into the end zone Superman style on a 2-yard run with about 8 minutes left.
He got going early when he shed one defender and hurdled another on a 35-yard run on his second carry. He finished that drive with a 2-yard touchdown run to give LSU a 7-0 lead. Fournette celebrated the score with the ''running off on the da plug twice'' dance from a song by rapper Plies. The dance basically looks like a spirited run in place, with arms pumping and a bit of a shuffle.
Texas Tech closed the gap when Grant grabbed a 46-yard touchdown pass, but the 2-point conversion failed and the score was 7-6.
D.J. Chark ran 79 yards for a touchdown on LSU's next play to make it 14-6. One Texas Tech defender caught up to him at about the 10-yard line but Chark simply dragged him into the end zone for the longest run in Texas Bowl history.
Fournette's fast start and the run by Chark left LSU with 143 yards rushing in the first quarter to minus-15 yards by the Red Raiders.
Fournette pushed LSU's lead to 21-6 when he took a screen pass 44 yards for a touchdown with about 6 1/2 minutes left in the second quarter.
The Red Raiders got within eight when Grant, who finished with 125 yards receiving, grabbed a 3-yard pass from Mahomes about 3 minutes later.
AP College Football website: collegefootball.ap.org
AP Sports Writer Kristie Rieken contributed to this report.