Alex Nicholls had opened the scoring for the Saddlers but they went down to 10 men when Weston was sent off in the 35th minute for two bookable offences.
Bobby Grant then equalised for the hosts shortly after the restart before veteran Paul Mullin scored a last-gasp winner to send Stanley into the second round.
Hutchings said: "The turning point was the second yellow card for Rhys Weston. I don't have any complaints but I think the tackle looked worse than it was.
"We then lost concentration for the last goal and you can't do that in any level of football. We let them get a cross in and the lad had plenty of time and space to score and we should have done better."