Carmichael's two free throws tied it at 75 with 2 seconds left in regulation.
Drexel led by three early in the extra period before the Redbirds (2-0) scored seven consecutive points. Brown hit a tying 3-pointer, then another basket for an 81-79 lead.
The Dragons (0-2) then turned the ball over and Johnny Hill's layup put the Redbirds ahead by four.
Drexel used a 12-1 second-half run to turn a three-point deficit into an eight-point lead. However, a 3-pointer by Brown, followed by a three-point play by Hill, got the Redbirds back into it at 71-69 with 5:06 left.
ESPN's Dan Dakich joins The Paul Finebaum Show to discuss the SEC's presence in the NBA Draft.
Ted Kapita was the top-rated recruit from West Virginia in the class of 2015 and recently played at DME Academy in Florida.
With the Suns drafting Tyler Ulis, they now have five guards on their roster who attended Kentucky.