The 6-foot-3 Florida native played in all 12 games for the Trojans last season and threw for a total of 1,669 yards, 14 touchdowns and six interceptions. Smith only started the final seven games, taking over for injured starter Kaleb Barker, and earned a record of 5-2.
Kentucky returns its starting quarterback, Terry Wilson, who played in all 13 games as a sophomore last season, throwing for 11 touchdowns and 1,889 yards. He also rushed for 547 yards and four scores.
As a graduate transfer, Smith will be able to play immediately with two years of eligibility remaining.
"We're excited to have Sawyer join our program," Kentucky coach Mark Stoops said in a statement. "It's great to add a quarterback with his experience and success."
Kentucky opens the season at home on Aug. 31 against Toledo.