Payton Hulsey got to the basket for a layup with under five minutes to play to pull Charleston (10-5, 2-2 CAA) within two, 50-48, but his foul in the backcourt on Drexel's next possession sent Tavon Allen to the free throw line, where he converted all three shots, then added two more less than a minute later to make it 55-48.
The Dragons (3-12, 1-3) with 11 of 13 from the line in the closing minutes to preserve the win.
Terrell Allen and Tavon Allen each contributed 11 points for Drexel.
Hulsey was the lone player to reach double-figure scoring for Charleston. The Cougars shot 43.5 percent from the field (20 of 46), including 7 of 15 from beyond the arc.