Pascal Siakam scored a career-high 21 points and the Aggies (6-9) were outshot 48 percent to 44.7 percent from the field but made 10 of 20 free throws.
The Tigers, behind Madarious Gibbs' 10 points, led 33-25 at halftime.
Baker hit two free throws and a 3-pointer and the Aggies pulled within one, 33-32, early in the second half. Baker's layup at 7:21 put the Aggies up 43-41, but Texas Southern's Jason Carter hit a jumper to tie it at 55 with 1:03 to play. The Aggies' Remi Barry missed a layup but DK Eldridge got the offensive rebound and New Mexico State called timeout with 22 seconds left. Baker made his game-winner on the ensuing possession.
Gibbs led the Tigers (3-10) with 15 points and Carter had 10.