Klein's 12-yard TD run in the third quarter proved to be the difference for the Wildcats (6-0, 3-0 Big 12) in what amounted to a tuneup for their meeting next week with West Virginia.
Klein finished the day 16-for-14 passing and ran for 105 yards on 25 carries.
Iowa State (4-2, 1-2) quarterback Jared Barnett was 16-of-36 for 165 yards and also led the Cyclones ground game with 34 yards on eight carries. Barnett threw touchdown passes to Shontrelle Johnson and Ernst Brun.
Jeff Woody scored on a 2-yard TD early in the fourth quarter to cut the Kansas State lead to 24-21 but Iowa State could not score again.
The Cyclones forced Kansas State to punt with just over 2 minutes to play but could not move the ball past their own 3-yard line.
Selena Gomez drops F-bomb, walks off stage during Jingle Ball performance
Trader Joe's: Car crashes into Long Island store, injuring 11