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.
SMU has signed a second player from Australia, 6-foot-11 forward Harry Froling, to a national letter of intent. The Mustangs previously signed guard Tom Wilson from Australia.
There's nothing to suggest the Jayhawks won't end up atop the Big 12 for a 13th season in a row. But what about a return to the Final Four?
Colorado State graduate transfer John Gillon has chosen Syracuse, giving the Orange a potential starter at point guard entering the 2016-17 season.