The Golden Eagles (7-1, 3-1 Conference USA), who won their sixth straight game and are knocking on the door of a Top 25 ranking, broke open a 10-10 game in the second half when Davis threw scoring passes of 47 yards to Quentin Pierce and 15 yards to Dominique Sullivan, and Woodyard scored on a 68-yard run.
Davis also had a 1-yard TD pass to Ryan Balentine in the first quarter and finished with 190 yards passing.
UTEP (4-4, 1-3), which was held to 262 yards of offense, got its only touchdown in the first quarter on a 1-yard run by Leilyon Myers. Dakota Warren added a pair of field goals for the Miners.
Joey Counts scored on a 1-yard run on fourth-and-goal with 2 seconds left to give Montana a 38-35 victory over four-time defending FCS champion North Dakota State on Saturday.
Transfer Vernon Adams will be the starting quarterback when Oregon begins its season against Eastern Washington on Sept. 5.
Tim Beckman is threatening legal action after Illinois fired him as football coach Friday amid allegations of influencing medical decisions and pressuring players to play hurt.
LSU offensive coordinator Cam Cameron revealed in a post-practice news conference that he has been treated for prostate cancer but plans to coach this season.
The woman who accused former Baylor defensive end Sam Ukwuachu of raping her in 2013 has hired a Title IX law firm to represent her in potential civil litigation.
Arkansas tight end C.J. O'Grady has been arrested by Fayetteville police and charged with driving while intoxicated and being a minor in possession of alcohol.