Luckett made 1 of 2 from the line, and then grabbed the offensive rebound off his miss to preserve the win.
Alcorn State (3-13, 2-2 Southwestern Conference) raced out after intermission and built an 11-point lead just 4 1/2 minutes into the second. A'Torri Shine's 3-pointer cut Grambling's deficit to eight with 9:45 left and Carl Lowe's 3 made it a three-point game with eight seconds left.
Alcorn State led 32-29 at the break. Thomas finished with 10 points and seven boards.
Shine led Grambling State (2-12, 0-4) with 21 points and Cormier scored 12.