It was all Houston early, as the Cougars used a 68-yard punt return score by Nathaniel Dell and Clayton Tune's 27-yard touchdown pass to Matthew Golden to lead 14-3 after one quarter.
Tulsa's Jack Long and Houston's Kyle Ramsey traded field goals before Braxton hit Keylon Stokes for a 1-yard score to get Tulsa (5-7, 3-5 American Athletic Conference) within 17-13 with 3:58 left before halftime. From there, Tune connected with Dell for a 13-yard touchdown, but Braxton answered with a 37-yard touchdown run and the Golden Hurricane trailed 24-20 at intermission.
Braxton and Santana teamed up for a 52-yard touchdown just two minutes into the third quarter to give Tulsa a 27-24 lead. Ramsey's second field goal tied the game at 27 after three quarters.
Braxton gave Tulsa the lead for good when he hooked up with Santana for a 25-yard score midway through the fourth quarter.
Braxton completed 25 of 43 passes and carried 11 times for a team-high 51 yards. Santana had six receptions for a career-high 169 yards.
Tune completed 26 of 47 passes for 386 yards with one interception for Houston (7-5, 5-3). Dell finished with nine catches for 161 yards.
Tulsa ended with a victory for the fifth straight season.