Caretaker managers Dean Glover and Andy Porter are still searching for their first point after seeing Port Vale outclassed for much of the afternoon.
County got off to the perfect start when they took the lead in their first serious attack in the sixth minute.
Delroy Facey headed Matt Hamshaw's far post cross back into the danger area where Jamie Forrester headed home from point blank range.
Vale's confidence levels were clearly very low and Forrester was close to adding a second before Anyon just beat him to a bouncing ball.
County were in total command and should have had the points wrapped up inside the opening half hour.
Facey missed an open goal from four yards and he, Forrester and Hamshaw all drew fine saves from the overworked Anyon.
The one-way traffic continued until the interval with Forrester again and Richard Butcher going close to an elusive second goal.
County were made for their profligacy three minutes after the interval when Vale equalised.
Shane Tudor delivered a right-footed cross from the left wing and Luke Prosser prodded a close-range shot past Russell Hoult at the far post.
The visitors then survived a period of pressure before scoring what proved the winning goal in the 67th minute.
Myles Weston exchanged passes with Forrester after a short corner on the left and the winger finished with a fierce shot, low inside Anyon's right-hand post.
Vale huffed and puffed after that, but they never seriously threatened a second equaliser and Facey came close to adding a third in the closing stages.