Tabb was 11 of 20 from the field and 3 of 9 from behind the arc for Alabama A&M (4-0), which continued its best start to a season in the program's D-I history. West had seven rebounds and Battle added five assists.
The Bulldogs led by as many as 13 points in the second half, but 11 straight points from Jacksonville State (2-5) made it 54-52 with 9:56 to play.
JaQuail Townser's jumper with 5:47 left tied it at 60-all, but the Gamecocks couldn't get over the hump. West's layup and 3-point play put A&M back on top and Battle scored six points in the final 0:39 to hold the lead.
Malcolm Drumwright had 18 points and five rebounds for Jacksonville State.