Barbour also had six assists for the Lions (7-5), giving the senior guard 301 for his career and tying Mike Jelinsky (1989-93) on the school's all-time list.
Steve Frankoski's 3-pointer with 4:41 remaining put Columbia ahead to stay, 56-54. The Lions took a 60-56 lead on Mark Cisco's jumper, with an assist from Barbour, and Barbour's jumper made it 62-56 with 2:26 remaining.
Columbia led 30-22 with 1:49 left in the first half, but the Raiders (5-10) went on a 14-0 run, spanning halftime, to go ahead 36-30.
Frankoski finished with 11 points for Columbia, which turned the ball over just seven times.
Mike Brey is looking at former Notre Dame players to fill two assistant jobs that have opened up with the departures of Martin Ingelsby and Anthony Solomon.
Purdue's 2016-17 prospects got a jolt Wednesday when forward Caleb Swanigan announced he will return for his sophomore season.
Clemson's Jaron Blossomgame says he is withdrawing from the NBA draft and returning to school for a final season.