Mullins scored 12 points in the first 10 minutes of the second half as the Lions (5-4) extended a 20-16 halftime lead to 42-27.
The Eagles (4-6) got within six, 44-38, on John Schoof's 3-pointer with 3:55 left, but Barbour scored nine points down the stretch.
Barbour became the 25th Columbia player to reach 1,000 career points and now has 1,016.
Stephen Lumpkins scored 18 points and grabbed 12 rebounds for American.
Both teams made just six field goals in the first half and the Lions shot just 20.7 percent (6 of 29), but nine American turnovers helped Columbia erase an early 9-1 deficit.
Columbia improved to 55.0 percent (11 of 20) in the second half. American shot 31 percent (13 of 42) overall.
Building a winning program is hard. Will sustaining that success prove to be even harder for Texas A&M?
Charles Matthews, who spent his freshman season at Kentucky, has picked Michigan as his transfer school, according to a source.
Chad Ford and Jeff Goodman offer their opinions on the surprises they're expecting at the NBA draft, with Ford believing there will be a lot of trade activity while Goodman thinks people will be surprised by potential draft spot of Marquese Chriss.