Rosenbauer had two short scoring runs before Robert Grebenc, who finished with 132 yards on 26 carries, added a 1-yard plunge to give the Bears (5-5, 3-4 Ivy League) a 21-0 lead in the first quarter.
Tuner DeMuth got Columbia (0-10, 0-7) on the board with 1:05 left in the fourth quarter with a 4-yard run.
Columbia was out-scored 389-103 as its second-straight winless season came to a close. This is the second-longest losing streak in Ivy history behind the Lions' 44-game run of futility in the 1980's.
Brown had 483 yards, 274 on the ground. Columbia had 228 yards, 26 rushing.