Qualls hit four 3-pointers, and Perkins had 14 points and 12 rebounds as the Tigers (5-13, 3-7 Southwestern Athletic Conference) ended a five-game losing streak.
Grambling used a 16-5 run sandwiched around halftime to take a 43-29 lead with 16:55 remaining. Qualls, Perkins, Rupert Rose and Greg Dillon hit 3-pointers during the run.
Southern (4-18, 2-8) cut the deficit to 58-55 on Tony McGilveary's free throw with 58 seconds to play, but Qualls and Perkins each hit three free throws to push the lead to 64-55 with 21 seconds left.
Madut Bol scored 11 points for Southern, which is 1-7 in its past eight games.
The teams split the season series, with Southern winning 73-67 on Jan. 9.