A back-and-forth game featuring mostly ugly basketball went down to the wire.
Charleston's Willis Hall plucked a deflected shot out of the air and hit a turnaround jumper with 3.8 seconds to play to give the Cougars the lead.
Drexel's Chris Fouch drove the length of the floor but missed a layup at the buzzer.
"I thought there might have been a goaltend, but it wasn't called," Flint said. "But we had a chance to distance ourselves earlier in the game, and we never did. We couldn't make a free throw. Free-throw shooting was the difference.
"I'm not going to make excuses about a long bus ride or anything like that. We could have won this game, plain and simple."
Drexel (13-11, 5-6 CAA) shot 40 percent from the field and was 0 for 9 from three-point range and 12 of 26 from the free-throw line.
The Cougars (13-13, 5-6 CAA) shot 32.7 percent from the field, including 2 for 13 from three-point range, and made 13 of 24 free throws.
Frantz Massenat led the Dragons with 16 points, while Tavon Allen added 12.
Drexel battled toe-to-toe with the Cougars through a sloppy first half, with Charleston's 24-18 halftime lead forged largely on some late theatrics from Canyon Barry, son of former NBA great Rick Barry.
With Charleston leading 20-18, Barry leaned in on a three-point attempt for the Cougars, and although he missed the shot, he drew enough contact to get the foul call. He made 2 of 3 from the line and, in the process, drew some commentary from the Drexel bench that resulted in a technical.
Barry made two more free throws for the 24-18 halftime lead.