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.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.