WACO, Texas -- Oklahoma had all the answers it needed to win a Big 12 College Football Playoff elimination game at Baylor.

Sooners running back Samaje Perine rushed for 166 yards and two touchdowns and quarterback Baker Mayfield threw a crucial fourth-quarter touchdown pass to lead No. 12 Oklahoma to a 44-34 victory over No. 6 Baylor on Saturday night at McLane Stadium.

In doing so, Oklahoma handed the Bears their first loss this season and ended the nation's longest FBS home winning streak at 20 games.

Not only that, the Sooners helped their CFP chances immensely, especially with no fewer than three teams ranked ahead of them losing on Saturday. Oklahoma coach Bob Stoops realized the importance of the win for his team.

"You go into the No. 6 team and into their backyard on a night like tonight and play this way, we're one of those teams that has a chance," Stoops said. "We're going to keep trying to improve and give ourselves an opportunity next week to play as well or better."

The Sooners have games remaining against No. 15 TCU and No. 8 Oklahoma State.

Baylor, which travels to Oklahoma State next week, took a huge hit to its chances of making it into college football's final four. However, Briles said his team will take its best shot as it attempts to win a third straight Big 12 championship.

"The good thing is we control our own destiny," Briles said. "Oklahoma has a Big 12 loss already. If we finish it out right, there could be a couple of us banging around with one loss and you never know what will happen."

Mayfield made the difference for Oklahoma in the closing minutes of the battle on the Brazos River, which runs past McLane Stadium.

Facing a third-and-goal from the Baylor 7 in the fourth quarter, Mayfield scrambled to buy time and found Dimitri Flowers open in the end zone for a 7-yard touchdown pass, extending Oklahoma's lead to 10 points with 4:47 left.

Perine's key run came midway through the third quarter with Oklahoma (9-1, 6-1 Big 12) already leading by seven. Perine found a seam in the middle and outran Baylor safety Orion Stewart to the end zone for a 55-yard touchdown.

"That's a big boost when you see the big man rolling down and outrunning everybody for a touchdown," Stoops said. "That charges everybody up."

In addition to Mayfield and Perine, Oklahoma wide receiver Sterling Shepard caught 14 passes for 177 yards, consistently keeping Sooners drives moving.

"Sterling again is just a sensational player," Stoops said. "Oh geez, 177, that's a good night. He's a special player. The guy's dynamic and makes so many competitive, tough catches."

But Baylor (8-1, 5-1) battled back as quarterback Jarrett Stidham threw two touchdown passes to shrink Oklahoma's second-half lead. He hit tight end LaQuan McGowan for a 5-yard touchdown and Jay Lee for a 17-yard score to cut the Sooners' lead to 37-34 with 10:29 left in the fourth quarter.

That's when Oklahoma put together a drive to take control.

The Sooners marched 78 yards in 10 plays, using up 5:42 of the fourth quarter. Mayfield capped the drive with the toss to Flowers.

Oklahoma safety Ahmad Thomas intercepted Stidham's pass on the second play of the ensuing possession to help ice the game.

"We didn't complement ourselves tonight," Briles said. "When the defense would get a big stop, the offense wouldn't do anything with the ball, and sometimes a little bit of vice versa."

Stidham, a freshman in his second start for the Bears, passed for 257 yards with two touchdowns and two interceptions. But he couldn't get the ball to Bears playmaker Corey Coleman, who was held to three catches for 51 yards.

Stoops said the Oklahoma defense passed the Coleman test.

"I thought (Jordan Thomas) did an awesome job out on (Coleman) when he was one-on-one," Stoops said. "We were very aware of him -- when we could help underneath him, when we could double him. Other times, you just had to live with the one-on-one matchup and I thought JT challenged him all night."

Mayfield finished with 270 passing yards with three touchdowns and an interception.

Mayfield threw a 39-yard touchdown pass to Shepard to put the Sooners ahead 13-6 midway through the second quarter.

Mayfield, though, handed the momentum back over to Baylor later in the quarter.

The Oklahoma defense forced Baylor to go three-and-out on two straight possessions in the second quarter. After the second, the Sooners had a chance to build momentum by driving for a two-score lead. However, Mayfield's pass on first-and-10 from the Oklahoma 42 was intercepted by Baylor nickel back Travon Blanchard.

The takeaway gave Baylor the ball in Sooners territory and set up a Bears touchdown drive. After a personal foul against the Oklahoma defense for a late hit on Stidham extended the drive on third-and-5 from the 6, Bears running back Devin Chafin ran 3 yards for a touchdown to make it 13-13 with 2:58 left in the half.

Mayfield made up for his pick by guiding Oklahoma on a 71-yard touchdown drive at the end of the second quarter. He completed all four passes on the drive for 61 yards. Mayfield scored the touchdown by running around the left end for a 2-yard score, putting Oklahoma on top 20-13 at halftime.

NOTES: The Bears last lost in Waco to TCU on Oct. 13, 2012. ... Baylor had never defeated Oklahoma going into the 2011 season. But the Bears, led by eventual Heisman Trophy winner Robert Griffin III, captured a 45-38 victory over Oklahoma in 2011 and have defeated the Sooners twice since. Oklahoma leads the series 22-3. ... Since losing to Texas 21-14, Oklahoma had beaten five straight opponents by an average of 38.4 points per game.