Strikes from Colby Bishop and Dion Charles looked to have secured the first win of the season for Stanley, but three late Shrewsbury Town goals ensured that the visitors left the Wham Stadium with all three points.
After a quiet first half, Bishop fired a penalty past Max O'Leary before Charles rocketed a shot into the far corner to give Stanley a 2-0 lead.
However, a Mark Hughes own goal and goals from Fejiri Okenabirhie and Daniel Udoh ensured that The Shrews completed a remarkable comeback.