Jamie Harris added 16 points for Drexel (4-6).
Tasheed Carr scored 19 points and Ahmad Nivins 16 for Saint Joseph's (5-7).
Leading 35-30 with 15:48 left, the Dragons went on an 11-4 run, then, after a free throw by Carr, scored 12 of the next 14 points to lead 58-37 at the 3:58 mark.
Saint Joseph's went on a 15-1 run to close within 59-52 with 32 seconds left, but could get no closer.
The Hawks used an early 10-0 run to take a 20-9 advantage, and led 26-16 with four minutes left in the first half.
Harris then sparked a 16-2 run and hit a 30-footer at the halftime buzzer that gave Drexel a 30-28 advantage at the break.