Klein rushed for 83 rushing yards on 12 carries and added 233 more on 19-of-26 passing as Kansas State (8-0, 5-0 Big 12) broke open a close game with 42 second-half points.
John Hubert scored twice en route to 64 yards, while Chris Harper made five grabs for 99 yards for the winners.
Seth Doege completed 34-of-49 for 309 yards with two touchdowns and an interception but it wasn't enough for the Red Raiders (6-2, 3-2).
Eric Ward picked up 146 yards and scored twice as Texas Tech saw its two-game winning streak snapped.
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