Jermain Easter opened the scoring for the League One side from the penalty spot after Craig Gardner was adjudged to have handled a Luke Chadwick cross into the box.
Five minutes later James McFadden missed the chance to level when he hit the crossbar from his spot-kick after Gardner was brought down in the box.
Half-time substitute Garry O'Connor hit the post with a header for Blues before the Dons doubled their advantage with Jabo Ibehre slotting home in the 47th minute.
But Jerome pulled a goal back in the 52nd minute after beating the offside trap and firing past Dons goalkeeper David Martin and he netted again from the edge of the box after slack defending 13 minutes from time.
O'Connor completed the comeback with two minutes left when his shot was deflected past Martin.