Blackpool remain three points outside the play-off zone after this defeat by their Hallmark Security League promotion rivals on a wet Monday evening in Shropshire.
Manager Dave Worthington could name only two substitutes with several players unavailable, and Billy McKenna and Ben Fishwick suspended.
And both subs were called upon early as club captain Martin Baird replaced Dan Mottram, who had a hamstring injury, only to suffer a similar injury himself within a few minutes and make way for Jack Taylor.
With no further substitutes available for the rest of the game in the torrential rain, AFC tried to play good football as Kit Gregory and Jamie Nay combined well down the left and Carnan Dawson got some decent balls into the box.
AFC came out stronger in the second half, with plenty of pressure early on, and an Alex Caunce free-kick was palmed round the post by the Alport keeper.
However, the home side then started to dominate, hitting the post on the hour and forcing Callum Kirkland into some fine saves.
They scored the winner in the 77th minute, when a shot rebounded around the area and Lee Pickerill pounced to score.
AFC could not force an equaliser and remain in eighth place in the first division with eight to play, while Whitchurch climb to third.
Blackpool are at home to fifth-placed Sandbach in a key contest on Saturday.
Blackpool: Kirkland, Nay, Dawson, Caunce, Mottram, Pickering, Bishop, Hargreaves, Duffield, Williams, Gregory; subs: Baird, Taylor.