Boro's unbeaten home record looked under threat at the break as they were a goal down and reduced to 10 men due to Michael Warner's red card, but second-half goals from Jon Nurse and Dino Maamria from the penalty spot and a Robbie Williams own goal meant an amazing transformation.
The Boro defence looked vulnerable in the early stages as Stanley pressed.
Right-back Warner received his first caution on 16 minutes and saw red 10 minutes later in the incident which lead to Stanley's goal.
He cynically halted the run of Gary Roberts to be ordered off and insult was added to injury as Steve Jagielka's scuffed effort went in after the initial free-kick was blocked.
Stanley were inches away from grabbing a second a minute before the break as Ronnie Henry scrambled clear off his own line.
Boro were rarely seen as an attacking force in the first half but this changed after the break despite their numerical disadvantage.
Nurse levelled on 52 minutes after George Boyd's shot was pushed out.
Six minutes later Nurse won a penalty as Leam Richardson pushed him over and Maamria scored from the spot.
After Stanley had an equaliser ruled out for offside, the unlikely three points were wrapped up 10 minutes from time when Williams sliced into his own net.