Hawthorne scored 16 of his 23 in the second half as the Dragons (9-9, 2-4 Colonial Athletic Association) built a 15-point lead before fending off the Panthers (4-12, 1-5).
Leading scorer Leonard Mendez (16.7 points per game) finished with just six on 2-of-11 shooting.
Hawthorne's 3-pointer with 8:01 remaining gave Drexel a seemingly comfortable 52-37 advantage. But Georgia State ripped off a 14-2 run to close within 54-51 with 2:50 left.
Chase scored six points during the spurt. The Panthers had numerous chances to get closer but misfired on four shots in the final minute. Elegar hit a layup and Hawthorne converted two free throws with 15.5 seconds remaining to seal the victory.
Drexel was outrebounded 37-29 but shot 52 percent from the field in the second half and never trailed in the game. The Dragons bench outscored Georgia State, 29-16.
Drexel improved to 5-1 all-time against the Panthers, which includes three consecutive wins.