In truth, Vale were fortunate to be two goals up at the interval as Cooper's vibrant side had been so dominant.
But Swindon were terrific after the break as they scored three times, struck the woodwork twice and created a string of other chances.
Cooper said: "The players were fantastic and I can't praise them enough.
"I didn't think we had done a lot wrong in the first-half, but we found ourselves two down and up against it.
"But I told the players at half-time to keep going to see if we could get an early goal back and go from there. That's exactly what happened and we were very worthy winners.
"We've got a lot of young players so there was no ranting and raving at half-time. I told them to keep believing and keep playing football because they are talented lads.
"They all did well but Alex Pritchard showed real quality all through the game. It was a brilliant free-kick for his goal and I've told him he should be doing that more often.
"We knew the conditions weren't going to be brilliant so we put Nile Ranger and Michael Smith up front to give us a target to aim at. They were a handful all afternoon and I'm sure the Vale defenders will tell you they had a tough game against them."
Swindon dominated much of the early play but Vale took the lead with their first serious attack in the 20th minute.
Jennison Myrie-Williams beat Darren Ward to deliver a low cross from the right and Doug Loft's right-foot shot on the turn from six yards out squeezed inside the near post.
The Robins' 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 visitors were back in the contest six minutes after the interval when Pritchard's cross from the right was headed into the roof of the net by Ranger at the far post.
Swindon then struck the woodwork twice as Smith's shot rattled a post before Ranger's effort clipped the crossbar.
But Cooper's men almost inevitably drew level when man-of-the-match Pritchard 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.