Bembry added eight rebounds, three assists, three steals and two blocked shots. James Demery scored 12 points for St. Joe's (1-1).
Brown's layup with 1:30 left broke a tie at 47 and gave the Hawks the lead for good.
Drexel turned it over before Isaiah Miles made the first of two free throws to make it 50-47 with 29 seconds left. Damion Lee hit a pair of free throws for Drexel, which quickly fouled after the inbounds pass and Shavar Newkirk made both free throws with 11 seconds left to close the scoring.
Lee, who scored 23 points and grabbed seven rebounds, missed a 3-pointer in the closing seconds.
The teams combined to make just three field goals in the final 6 minutes.