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.
The family of former Houston Astros owner Roy Hofheinz wants a court to require the University of Houston to keep his name on the school's basketball arena.
Michigan has hired assistant basketball coaches Billy Donlon and Saddi Washington.
Duke and Cincinnati, 2016 NCAA tournament teams, highlight the eight-team field for the Naismith Memorial Basketball Hall of Fame Tip-Off Tournament.