"It's pretty exhilarating right now. I'm so proud of this team," Webb said.
"When Coach told me I was going to start, there was nothing but joy," Webb said. "I told myself that I was going to make sure that I was going to send the seniors out right. We came prepared. We leaned on each other and we showed that today."
The Red Raiders (8-5) won for the first time since beating West Virginia on Oct. 19. Arizona State (10-4) lost its second straight.
Webb completed 28 of 41 passes. He threw touchdown passes of 18 and 21 yards to Jakeem Grant, 1 yard to Rodney Hall and 23 yards to Bradley Marquez, all in the first half.
"He's really battled to separate himself all year," said first-year coach Kliff Kingsbury, a former Red Raiders quarterback.
"To play that young in this type of arena, D-1, that's very rare and I think he does a great job handling the pressure coming out here and making things happen," wide receiver Eric Ward said.
After losing their final five games by an average of 20.6 points, the Red Raiders scored on four of their first five drives to take a 27-6 lead 5 minutes into the second quarter.
Kingsbury said the Red Raiders had seen predictions that they'd lose by 22 points.
"We saw that all week. We knew what people thought and we knew the type of team we had," the coach said.
"We didn't get any respect all weekend long," Ward said. "It was all about Arizona State and their record and this, that and the other. So we used that to fuel the fire."
Ward said the losing streak "preyed upon us pretty heavily. No one likes losing that much and lose that much on the road. It starts depressing the team a little bit. We just did all we could to prepare for this game. During the bowl practices, the time we had together helped us bond together."
Texas Tech All-America tight end Jace Amaro had eight catches for 112 yards. He set the FBS single-season record for yards receiving by a tight end with 1,352.
The Sun Devils, coming off a 38-14 loss to Stanford in the Pac-12 championship game, were held 18 points below their average. They dropped to 0-4 in the Holiday Bowl dating to 1985.
"I'm very embarrassed for our program," coach Todd Graham said. "As coaches we didn't have our guys prepared to play. They came to play. We didn't. Give them credit. I was embarrassed for the fans that showed up today. We didn't have our players prepared. Zero excuses, but we didn't have our players prepared to play. It was an ugly game offensively and defensively. We just didn't play very well. They were the more passionate team today."
While Texas Tech was finding the end zone early on, Arizona State had to settle for field goals of 44 and 31 yards by Zane Gonzalez.
The Sun Devils got their first touchdown when D.J. Foster scored on a 20-yard run with 7:52 left in the second quarter to close to 27-13. ASU quarterback Taylor Kelly scored on a 44-yard run early in the third, but Reginald Davis returned the ensuing kickoff 90 yards for a touchdown to extend the Red Raiders' lead to 34-20.
Kelly gained 135 yards on 25 carries but threw for only 125 yards, going 16 of 29. He was sacked three times. Foster gained 132 yards on 20 carries.
Arizona State's Richard Smith couldn't hold onto a 50-yard pass from Kelly as he tripped and tumbled into the end zone with 9:50 left and the Sun Devils trailing by 14.