Bradford broke former Heisman runner-up Josh Heupel's school record with the 14th game of his career with at least 300 yards passing, and he could have had even bigger numbers if the Sooners' receivers didn't have trouble handling his passes.
Bradford had a career-high 22 incompletions, but at least six of those came on drops. A few other passes also hit off receivers' hands.
Bradford had been out for a month with a sprained AC joint in his right, throwing shoulder and had hoped to return this week so he'd have a game under his belt before next week's rivalry game against No. 2 Texas.
Brown's touchdown runs of 1 and 4 yards in the second quarter put Oklahoma up 14-0, but the Sooners (3-2, 1-0 Big 12) couldn't get into the end zone on their first four trips inside the 20-yard line in the second half.
Jimmy Stevens hit four field goals to tie a school record held by Jeremy Alexander and Garrett Hartley before Bradford finally connected with Adron Tennell in the back of the end zone for a 3-yard score with 4:03 to play.
There had been concern that Bradford's shoulder wouldn't have the stamina to hold up to a full game, but he ended up matching the second-most passes he'd thrown in his career with a 27-for-49 performance. The only time he had attempted more passes was when he threw 53 against Kansas last season. He had never had more than 19 incomplete passes in a game.
Bradford skipped part of Oklahoma's pregame routine while the rest of the team's quarterbacks were practicing throwing, and he then jogged to the locker room while his teammates were stretching.
Then came a real scare when trainers huddled around Bradford after he stepped awkwardly on an incomplete pass on his first drive. Bradford appeared to be testing out his right knee on the sideline, but never missed a play until he was taken out late in the fourth quarter with the game in hand.
Brandon Caleb caught seven passes for a career-best 139 yards after moving inside to replace injured slot receiver Ryan Broyles, who is out 2-4 weeks with a hairline fracture of his left shoulder blade. DeMarco Murray added 113 yards rushing on a season-high 24 carries.
Third-stringer Nick Florence started at quarterback for Baylor (3-2, 0-1) for the second straight week, following a season-ending knee injury to Robert Griffin III. He threw for 220 yards on 22-for-41 passing with one touchdown and one interception.
Blake Szymanski, the backup who had sat out last week with a bruised shoulder, replaced him on Baylor's final series in the fourth quarter and threw a 42-yard completion and an interception in the back of the end zone on consecutive plays.
Jay Finley, the Bears' top rusher, also returned after missing two games with an ankle injury but had just two carries for minus-1 yard.
The Sooners have won all 19 games in their series against Baylor and won for the 27th straight time at home.