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.
College basketball coaches across the country will hit the recruiting trail hard in July to target the Class of 2017's top prospects. Which players are the nation's top programs targeting?
With only a handful of players expected to return after next season, John Calipari has Kentucky in the mix for nearly every elite 2017 prospect.
If Arkansas coach Mike Anderson wants to go back to the NCAA tournament and get off the hot seat, he'll need a big year from Moses Kingsley.