Barkley completed 23-of-38 passes for 377 yards. Lee grabbed a scoring strike and also returned a kickoff 100 yards for a touchdown for the Trojans (1-0).
The pair hooked up on a 75-yard score in the game's first play from scrimmage.
USC was led on the ground by Silas Redd, who gained 57 yards on nine carries in his first game since transferring from Penn State.
The Warriors (0-1), playing their first game as a member of the Mountain West Conference, were led by Sean Schroeder, who completed 23-of-38 passes for 208 yards with a TD and two interceptions.