Ben Malena (7 yards) and Trey Williams (30 yards) had the TD runs to cap the long scoring drives in the third quarter for the Aggies.
Manziel was picked off in the second quarter after his bootleg move and a throw that hit Malcome Kennedy in the hands in the end zone and deflected into the air. Javon Harris grabbed the interception.
The Sooners then crossed midfield before Jones had a pass intercepted by Dustin Harris and returned to the Oklahoma 48.
That A&M drive started with a little trickery, Manziel holding the ball down in his left hand while faking a throw with his right hand. He then pitched to Kenric McNeal, who threw a 20-yard pass to Mike Evans. Two plays later, Manziel had his bootleg TD run.
Oklahoma was in the Cotton Bowl for only the second time. It was the first bowl matchup between the former Big 12 rivals, but the 17th consecutive season they have played each other.
The Sooners had won 11 of 13 since Bob Stoops became their coach. That included a 77-0 Oklahoma win in 2003 that was the most-lopsided loss in Texas A&M history.
Sumlin was the A&M offensive coordinator in 2002 when the Aggies upset the top-ranked Sooners. The next year, Sumlin was hired by Stoops as an assistant, and he stayed there five seasons before going to Houston as head coach and then the Aggies.
"Words can't describe how I feel," said Damontre Moore, A&M's leading tackler who has already said he will bypass his senior season for the NFL draft. "I'm just overwhelmed with excitement and joy, just to get such a big win, all the goals that we set for ourselves at the beginning of the season, to see them be accomplished."