
The Scholars came from behind to secure three points on the road in Shropshire.
There were three changes to the side that had narrowly lost to Bootle, with new signing Kieron Dawes making his debut and Jack Langston and Joey Butlin returning to the starting line-up.
Neither keeper was tested until Langston's 35th-minute free-kick was pushed wide by Wycherley.
Nearing half-time, Sam Wilding's diagonal ball found George Cater, whose shot towards the far post was kept out by the outstretched Shifnal stopper.
Kieran Cook put the hosts ahead on the hour. A ricocheted ball landed in the former Scholar's path, and he battled through to put the ball in the net.
Chasetown were level ten minutes later, with Cater leaving the Shifnal keeper flat-footed with a looping strike from the corner of the box.
The momentum was with the visitors, and the match-winning blow came in the 83rd minute. Wilding's direct ball forward was helped on by Tyrell Skeen-Hamilton and pounced upon by Butlin, who slotted past the keeper to the delight of the travelling fans.