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.
Reggie Witherspoon, who spent more than 13 seasons coaching at Buffalo, is returning to the city to coach Canisius College.
North Carolina Hall of Fame men's basketball coach Roy Williams is recovering from knee-replacement surgery.
Sparkling facilities are one thing, but Tim Miles needs to regain the momentum of Nebraska's 2014 run to the NCAAs if it is to play with the big boys.