All-SEC quarterback Tyler Wilson threw two touchdown passes, including a 9-yarder to Kobi Hamilton with 4 minutes left in the third quarter after Kansas State (10-3) scored 16 consecutive points.
Adams' 51-yard return early in the second quarter gave the Razorbacks (11-2) a 10-0 lead before they even gained a first down. It was the first punt return for a touchdown in the Cotton Bowl in 51 years.
Arkansas' only losses were to No. 1 LSU and No. 2 Alabama, the SEC West rivals who will play in the BCS national championship game.
Kansas State quarterback Collin Klein tied the Big 12 record with his 27th rushing touchdown this season and matched the FBS record for quarterbacks.
After being held to 15 yards on 12 carries in the first half, Klein finished the opening drive of the third quarter with a 6-yard TD run that capped 16 consecutive points for Wildcats to get them to 19-16. That came after Klein had a 15-yard run on the previous play.
That matched the Big 12 record for rushing touchdowns held by 1998 Heisman Trophy winner Ricky Williams of Texas. The only other FBS quarterback with 27 rushing touchdowns in a season was Ricky Dobbs for Navy in 2009.
But the Razorbacks extended their lead on Wilson's 9-yard TD pass to Cobi Hamilton.
Adams' 51-yard return in the opening minute of the second quarter gave the Razorbacks a 10-0 lead before they even gained a first down.
Tyler Wilson threw a 45-yard TD pass to Jarius Wright with 4 minutes left in the half for a 19-0 lead before Kansas State blocked the extra point and Nigel Malone returned it for two points.
Klein, who rushed for 1,099 yards and 26 touchdowns during the regular season, threw a 3-yard TD pass to Andrew McDonald with 26 seconds left in the first half after Wilson was sacked by Adam Davis and fumbled at the Razorbacks 13.
Wildcats defensive end Meshak Williams was injured at the end of the first half, when he made helmet-to-helmet contact with Emmanuel Lamur when both were trying to tackle Wilson, who put his head down to avoid the hit.
Williams remained down on the field for several minutes being tended to by medical personnel and then placed on a stretcher. He gave a thumbs-up signal when being placed on a cart, then when he was being driving off the field extended his right arm high and flashed a Wildcat sign.
Kansas State officials said there was no immediate word of his specific injury or condition.
The Razorbacks played their second game at Cowboys Stadium this season, three months after they overcame an 18-point halftime deficit to beat Big 12 team and future SEC foe Texas A&M 42-38.
In their return trip, Arkansas led the Big 12 opponent at halftime and had mostly throttled Klein and the Kansas State offense. It was the first game for new Razorbacks defensive coordinator Paul Haynes, who was hired in December after Willy Robinson resigned.
Zach Hocker kicked a pair of field goals (26 and 22 yards) for Arkansas.
Wilson, who threw for a school-record 510 yards in that Oct. 1 game at Cowboys Stadium, was 16 of 26 for 163 yards after three quarters against Kansas State. Wright had two catches for 54 yards; he had 13 catches for a record 287 yards against the Aggies.
The punt return for a touchdown by Adams was the first in 51 years at the Cotton Bowl. The last was by also by a Razorback, when Lance Alworth had a 49-yard punt return for a touchdown in a 7-6 loss to Duke in the 1961 game.
Arkansas led 3-0 after Jake Bequette sacked Klein and forced a fumble, setting up Hocker's 26-yard field goal.
Tenarius Wright, the Razorbacks' other defensive end, jumped over a pile to recover the loose ball at the Kansas State 13.
Klein's fumble came on a second-and-33 play after a sequence that started with his 20-yard run to inside Arkansas territory being wiped out by an illegal shift penalty. He was sacked on the next play before a holding call and a false start pushed the ball all the way back to the 20.
The matchup of 10-2 teams is the only non-BCS game with both teams in the top 15 of the Bowl Championship Series standings - Arkansas is sixth, Kansas State eighth.
It also is the only bowl featuring the SEC vs. the Big 12, the top two conferences in the BCS computer rankings.
Arkansas' only regular-season losses were to No. 1 LSU and No. 2 Alabama, their SEC West rivals who will play for the BCS national championship Monday night.
At Kansas State, coach Bill Snyder has directed his second Wildcats turnaround. K-State could match the school record with its 11th victory, a mark it reached six times during a seven-year span in Snyder's first tenure before a three-year retirement.
Eyewitness Sports anchors Bruce Heartl and Jenn Bates are at Cowboys Stadium for the game. Look for their updates on KWCH.com and on Eyewitness Newscasts.