Mississippi State (4-7) opened with a 14-3 run and led 33-25 at the half.
But Alabama A&M (4-7) held the Bulldogs scoreless for the first 7 minutes, 22 seconds of the second half and eventually took a 35-33 lead.
The lead changed several times throughout the rest of the half, and with 43 seconds remaining, Green Hill hit a layup to give Alabama A&M a 58-57 lead. After a failed possession by Mississippi State, Hill hit another free throw to seal the win.
Tabb was 7 of 16 from the field and Crutcher was 5 of 10.
As a team, Alabama A&M shot 42.1 percent from the field (24 of 57) and 31.3 percent (5 of 16) from beyond the arc.
Brandon Ellis added 13 points for Alabama A&M along with seven rebounds and three blocks.
The Bulldogs shot 35.7 percent from the field (20 of 56) but hit only seven shots in the second half. Mississippi State was only 4 of 15 from beyond the arc.
Jalen Steele scored 10 points for Mississippi State in his first game back from a wrist injury on Nov. 13.
Alabama A&M is the second Southwestern Athletic Conference team to beat a Southeastern Conference team this season after Southern beat Texas A&M on Dec. 22.
adidas Men's Mississippi State Bulldogs Maroon Long Sleeve Shooter ShirtPrice: $47.97 Shop
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point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.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.