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.
College basketball fans should be excited about next season's Jimmy V Classic, an event that will feature Duke's heralded freshman class.
Jeff Goodman logs all the coaching changes following the 2015-16 college basketball season.
Can Josh Pastner keep top talent in Georgia? Will Tubby Smith change his recruiting philosophy at Memphis? New coaches face some difficult questions.