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.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.