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.
Florida State's student government has passed a resolution discouraging Seminoles fans from wearing Native American headdresses at games.
The second annual Gavitt Tipoff Games, eight games between members of the Big East and Big Ten to be played Nov. 14-18, will feature Villanova at Purdue in the opener.
Michael Lewis is leaving his assistant coaching job at Butler to take the same position at Nebraska under Tim Miles.