ARLINGTON, Texas — Billed as the Cowboys Showdown, the Texas-UCLA matchup played out like an old-fashioned Western shootout.

Both teams played it close to the vest, eyeing their opponent for three quarters. Then when they finally drew in the fourth quarter, UCLA had the better aim to claim a 20-17 victory at AT&T Stadium.

UCLA backup quarterback Jerry Neuheisel threw a 33-yard touchdown pass to junior wide receiver Jordan Payton as the Bruins took back the lead with three minutes left.

“What a heck of a game between two gutty teams,” UCLA coach Jim Mora said. “We were fortunate to get away with a win. We’ll take it and get on that plane real fast.”

Earlier in the fourth, Texas sophomore quarterback Tyrone Swoopes led the Longhorns (1-2) on a 10-play, 80-yard touchdown drive. Swoopes threw an 8-yard touchdown pass to John Harris to give Texas a 17-13 lead with 5:13 left.

UCLA punt returner Ishmael Adams set up the winning touchdown with a 45-yard punt return to the Texas 33 with 3:06 left. On the next play, Neuheisel hit Payton for the game-winner.

Swoopes had the ball in his hands again with 2:55 to go, but the Bruins (3-0) stopped Texas from even gaining a first down. Swoopes threw incomplete on fourth-and-7 from his 26-yard line. UCLA took over on downs and ran out the final 1:50.

Neuheisel, who played the final three quarters in place of injured starter Brett Hundley, finished with 178 passing yards and two touchdowns. Teammates lifted Neuheisel on their shoulders after the win.

Mora said he was enthusiastic knowing Neuheisel, the son of former UCLA coach Rick Neuheisel, was going in the game in the first quarter.

“My first thought was excitement for Jerry Neuheisel,” Mora said. “This kid is everything that’s right about college football and about UCLA. He bleeds Bruin blue. You couldn’t write a better script than this right here — for him to come to Texas and bring his team back for a win.”

Hundley left the game after running 11 yards for a first down with 4:29 left in the first quarter. He stood on the sideline with his left elbow wrapped during the second half. Mora said the elbow would be evaluated by the UCLA medical staff when the team returns to Los Angeles.

Neuheisel took UCLA the rest of the way on a drive that reached the Texas 29 before kicker Ka’imi Fairbairn booted a 47-yard field goal for a 3-0 lead.

Hundley remained on the sideline with his helmet off until UCLA went back on the field for its next possession. But Neuheisel returned to the game and Hundley went to the locker room.

Meanwhile, Swoopes completed his first 11 passes and looked more confident running the offense as the first half progressed. He finished the half completing 13 of 15 passes for 123 yards and a touchdown.

Swoopes completed two passes and running back Malcolm Brown ran for 22 yards as Texas drove to the UCLA 14 before Nick Rose kicking a tying field goal from 33 yards.

The Longhorns then took the lead as Swoopes heated up, connecting on 6 of 8 passes on a 14-play, 62-yard touchdown drive.

Swoopes kept the drive going with a 33-yard completion to senior wide receiver John Harris on fourth-and-8 from the UCLA 38. Swoopes capped the drive with a 2-yard touchdown pass to tight end M.J. McFarland that gave Texas a 10-3 edge that it took to halftime.

Swoopes completed 24 of 34 passes for 196 yards and two touchdowns.

“He just continues to get better and better,” Texas coach Charlie Strong said. “The team is all behind him and he’s big and strong enough to where he can make the throws. He’s just getting better and better, week by week.”

However, UCLA outgained the Longhorns by more than 100 yards, mostly on the ground. The Bruins rushed for 217 yards, led by Paul Perkins’ 126 yards on 24 attempts.

“We didn’t stop the run,” Strong said. “You can’t miss tackles. We missed a lot of them. We weren’t fitting our gaps and getting off blocks and we weren’t able to establish the running game.”

NOTES: Although the UCLA-Texas series was tied at three wins each entering the game, there haven’t been many close ones between the Bruins and Longhorns. Only one game — the first meeting in 1970, which Texas won 20-17 — had been decided by fewer than 18 points. Texas won the last meeting, 49-20 in 2011, to stop UCLA’s three-game winning streak in the series. Saturday’s game at AT&T Stadium was the first time that UCLA and Texas met on a neutral field. … Entering the game, UCLA had won nine of 10 nonconference matchups, with the only loss coming against Baylor in the 2012 Holiday Bowl. … It was only Texas’ second game at AT&T Stadium. The Longhorns defeated Nebraska 13-12 in the 2009 Big 12 Championship game.