Tech (8-6) appeared finished after Minnesota (6-7) took a 38-7 lead with 7:47 to go in the third quarter. But the Red Raiders mounted a furious comeback, scoring 31 unanswered points in less than 20 minutes.
Alex Trlica's 52-yard field goal as regulation expired sent the game into overtime.
Tech's comeback began with 4:58 to go in the third quarter, when Graham Harrell hit Joel Filani for a 43-yard score to cut the lead to 38-14. That touchdown started an avalanche that buried Minnesota.
Trailing 38-35 with no timeouts, the Red Raiders took over at their own 11 with 1:06 remaining. Eight plays later, Trlica tied it.
Harrell threw for 445 yards and two touchdowns. Woods rushed for 109 yards and three touchdowns and Filani caught nine passes for 144 yards.
Minnesota set a school bowl scoring record, and Cupito, a senior, tied Asad Abdul-Khaliq's career record of 55 touchdown passes. The records were little consolation in the end.
The Golden Gophers jumped ahead 7-0 after Texas Tech coach Mike Leach went for it on fourth-and-1 at his own 45. Harrell was stopped on a sneak, and six plays later Cupito found tight end Jack Simmons for a 2-yard touchdown with 9:27 to go in the first quarter.
Four minutes later, Minnesota made it 14-0 after linebacker Mike Sherels intercepted Harrell at Tech's 37. Pinnix capped a six-play drive with a 2-yard run.
Another Harrell turnover killed a Tech scoring drive. He fumbled on a sack by Willie VanDeSteeg, and Steve Davis recovered at the Golden Gophers' 13. Minnesota marched 87 yards -- its longest scoring drive of the year -- to take a 21-0 lead on Justin Valentine's 1-yard plunge on the first play of the second quarter.
Tech had a chance to slice the deficit when cornerback Antonio Huffman picked off Cupito's pass at the Minnesota 20. But Pinnix jarred the ball loose, and it bounded into the end zone, where the Gophers recovered for a touchback.
After Tech's Shannon Woods scored from 1 yard out to make it 28-7, the Gophers answered with an 81-yard drive that ended in a 3-yard touchdown pass from Cupito to Logan Payne in the final minute of the first half.
Minnesota looked as if it ended any Tech hopes for a comeback by opening the third quarter with a 16-play, 78-yard drive that consumed 7:13. Joel Monroe's 20-yard field goal gave the Gophers a 38-7 lead.
As it turned out, Texas Tech had plenty of time left.