The visitors dominated much of the early play but Vale took the lead with their first serious attack in the 20th minute.
Swindon's general play continued to impress as they forced eight first-half corners, but Tom Pope's first goal in six games left them two down at the break.
The unmarked Pope pounced on some dreadful defending as his left-foot shot from 10 yards squirmed just inside Wes Foderingham's left-hand post in the 42nd minute.
The Robins were back in the contest six minutes after the interval when Alex Pritchard's cross from the right was headed into the roof of the net by Nile Ranger at the far post.
The visitors then struck the woodwork twice as Michael Smith's shot rattled a post before Ranger's effort clipped the crossbar.
But Swindon almost inevitably drew level when man-of-the-match Prichard curled a 25-yard free-kick into the top right-hand corner of the net with a sumptuous right-foot shot after 65 minutes.
And a ragged Vale outfit could have no complaints four minutes later when Smith's low cross from the right was bundled in at the far post by Nathan Byrne.