Ash's 36-yard scoring strike with 2:24 left in the contest helped erase a 14-point, fourth-quarter deficit and gave the Longhorns (9-4) their first win in three games to end the season.
Ash was 21-of-33 passing for 241 yards, two scores and one interception. Goodwin finished with four catches for 68 yards and added a 64-yard touchdown run for Texas.
Jaxon Shipley led the Longhorns' receiving corps with eight grabs for 88 yards.
The Longhorns were playing without junior linebacker Jordan Hicks and junior backup quarterback Case McCoy, who were suspended Friday for violating team rules.
Reports indicated San Antonio police were investigating an alleged sexual assault of a 21-year-old woman at a city hotel.
Oregon State (9-4), meanwhile, lost in its first post-season appearance since the 2009 Las Vegas Bowl.
Cody Vaz was limited to 194 yards on 15-of-28 passing and was picked off twice for the Beavers.
Storm Woods rambled for 118 yards and two touchdowns on 21 carries for Oregon State.