Blackpool made it two win and two clean sheets in four days with victory at in-form Heys.
The Bury club are third in the Hallmark Security League first division and had scored nine goals in their previous two games.
AFC manager Dave Worthington made three changes to the side that won at Holker as Marc Taylor (pictured), Conah Bishop and Alex Taylor were recalled.
There were few chances in a cagey first half, in which AFC restructured after Jamie Nay replaced the injured Matt Cassidy.
The hosts had a spell of pressure early in the second half but AFC continued to hold their own, despite being forced into a second change when threatening winger Billy McKenna was forced off with a facial injury, Stephen Betteridge slotting in.
The winning goal arrived on 69 minutes, when Conah Bishop latched on to an excellent pass by left-back Conah Dawson to fire home.
AFC held on to their lead impressively to climb to 11th
Boss Worthington said: “I am delighted with the result against a strong Heys side who had scored the most goals in the division. It was fully deserved and we proved that we can be a match for any team in this division.
“The lads are still learning to play together as a team and this is our best result so far.”
Blackpool are at home to premier division West Didsbury & Chorlton in the FA Vase on Saturday.
Blackpool: Kirkland, Baird, Dawson, Cassidy, Vaughan, M Taylor, McKenna, Fishwick, A Taylor, Gregory, Bishop; subs: Betteridge, Duffield, Nay, Ngamba, Speight.