Brewer, who played in 13 games while throwing for 440 yards and five touchdowns in his Tech career, suffered a back injury during the summer that knocked him out of the preseason quarterback competition. Brewer returned to practice in October, but was never able to climb up the depth chart.
Walk-on Baker Mayfield, who won the starting job in the preseason and finished the regular season as the starter, already had left Texas Tech unexpectedly before the National University Holiday Bowl and has enrolled at Oklahoma.
Mayfield told ESPN's Joe Schad on Wednesday that he plans on staying at Oklahoma despite losing his appeal to not give up a year of eligibility.
Brewer intends to graduate from Texas Tech at the end of the spring semester and will be a graduate transfer eligible to play in 2014.
"I am honored to receive my degree from Texas Tech and will always call this university home," Brewer said in a statement. "... Wherever I may go, I hope our fans will respect the person and player that I am. The decision to transfer was well thought out by my family and myself and was one of the toughest of my life. I feel it's the right decision for me moving forward. As always, Guns Up!"
The Red Raiders will have only one scholarship quarterback this spring in Davis Webb, who as a true freshman took over the starting role last season after Mayfield suffered a knee injury during Week 5 and rallied Tech past West Virginia.
Webb also got the start in the Holiday Bowl and threw for 403 yards and four TDs in a 37-23 upset victory against Pac-12 South Division champion Arizona State.
ESPN's Joe Schad contributed to this report.