The win kept the Braves (3-17, 3-7) in contention for the eighth and final SWAC tournament spot. They also won two in a row for the first time this season.
The Tigers (12-10, 8-2), who fell a game behind Texas Southern, made only 26 of 41 free throws.
Alcorn State led most of the game, but Jackson State pushed ahead 42-40 early in the second. The Braves used a 16-3 run for an 11-point lead, but the Tigers tied it with an 11-0 spurt, including nine points from Grant Maxie.
A Baker 3-pointer gave Alcorn State the lead for good with 1:44 left.
Korey Sanders added 14 points and 13 rebounds for the Braves.
Jenirro Bush had 22 points and Tyrone Hanson 21 to pace the Tigers.