Sports Illustrated earlier reported that Rosen wanted to play, but the decision was "100 percent medical."
"I want to be clear on this: Josh wanted to play,'' Bruins interim coach Jedd Fisch said after the game. "Josh was unable to play because of the fact that he had two concussions within a four-week span in November, and our physicians didn't feel comfortable putting him out there and putting him at risk for a possibility of a third concussion.''
Rosen sustained his second concussion of the season in UCLA's regular-season-closing victory over California.
Freshman Devon Modster, who appeared in five regular season games, started for the Bruins. Modster finished the game 21 for 34 passing for 295 yards and two touchdowns.
ESPN's Josh Weinfuss and The Associated Press contributed to this report.