Four wins in a row had rekindled their early season ambitions when they topped League One, but this defeat followed hard on the heels of a 3-0 thrashing at Nottingham Forest.
Anyon, who has come up through the youth scheme at Vale Park, was called off the bench after two minutes when goalkeeper Mark Goodlad slipped on the wet surface and couldn't get up.
City, with problems of their own, had to come from behind though for the third match running. Scott Murray's second-half speed put them on top and he flashed in the left-footed winner to make it four goals in his last five matches.
It was ironic that on a day the ground staff worked hard to get the game on it didn't rain but it poured for the Valiants.
An hour before the kick-off they announced top-scorer Leon Constantine would miss out through injury prompting rumours of a transfer window move to Championship side Barnsley.
But there was no hint of anything but a genuine injury when Goodlad was stretchered off ten minutes later, while Colin Miles limped away with a sore calf which had been troubling him all week.
Anyon, who was given a run-out in the Johnstone's Paint Trophy two months ago, was soon in action grasping a Enoch Showunmi header as the striker turned on a left-wing cross from Jamie McAllister.
He followed this with a couple of confident dives but luck was with him too as Louis Carey hit the outside of his post and Steve Brooker incredibly headed wide of an open goal.
As Brooker reached for a Showunmi cross he had only to wait for the ball to hit him to score. The goal was gaping with the young keeper out of position and his team-mates could only look on in disbelief as the header went wide of the far post.
Meanwhile, Brooker was getting little change from former team-mate Clayton Fortune, who left Ashton Gate in the summer. The tall central defender, who has worked up a good understanding with skipper George Pilkington, had a great first half but a not so good second.
Vale were looking for a win when Jeff Smith flighted in a 68th minute free-kick, but within a minute Brooker had levelled with a low shot as the ball was flicked on to him.
It was all set up for Murray to give City their grandstand finish with the winner 14 minutes from time.