The Braves (7-11, 4-2 Southwestern Athletic Conference) shot 51 percent from the field. Twann Oakley had 16 points on 4-of-5 shooting from beyond the arc and dished out five assists.
UAPB (4-13, 2-3) was held to 17-of-44 shooting (38.6 percent) and committed 15 turnovers. Davon Haynes led the Lions with 22 points and 10 boards.
Alcorn State broke a 13-all tie as Luckett scored seven points to spark a 9-0 run, and the Braves coasted to a 33-20 halftime lead.
Haynes hit a layup in the second half to make it 40-27, but Alcorn State pulled away as Oakley hit two late 3-pointers to push the Braves' lead to 25.
Alcorn State has won four straight after dropping its first two conference matchups.
ESPN's Dan Dakich joins The Paul Finebaum Show to discuss the SEC's presence in the NBA Draft.
Ted Kapita was the top-rated recruit from West Virginia in the class of 2015 and recently played at DME Academy in Florida.
With the Suns drafting Tyler Ulis, they now have five guards on their roster who attended Kentucky.