Vale, who in League Two are currently sat in second place and only three points behind top spot, put in a performance that Darrell Clarke could be proud of as they looked defensively sound with Lucas Covolan having very little to deal with.
Going forward, they looked deadly on the counter-attack and took their chances when getting in behind.
Wilson opened the scoring inside half an hour with a composed finish before in the second period Colby Bishop was dismissed for an off the ball incident.
This seemed to change the game as minutes later Wilson's saw his deflected strike hit the back of the net before he tapped home with 11 minutes to go to confirm the match ball.
Story of the game
Stanley began the affair the better, dominating possession and they missed a gilt-edged opportunity to break the deadlock.
This was when Harry Pell slipped in John O'Sullivan down the right and his cross was spilt by Lucas Covolan into the path of Hamilton, but he lashed his strike well over the bar from around 12 yards.
Then just before the quarter of an hour mark after some sloppy play out the back by the hosts, Pell was found in space down the right but he could only see his shot flash narrowly past the far post in what was a massive warning to Clarke's side.
With the game settling down, the Valiants took the lead when from a throw-in from the right, David Amoo flicked the ball onto Wilson, he did brilliantly to turn away from the defender and when through in the box, he had the composure to slide the ball into the bottom left-hand corner.
With confidence now high and John Coleman's side looking shellshocked, Wilson had a chance to double his tally four minutes before the interval but his first-time effort from a clever low corner kick routine was sliced well over the bar.
On the stroke of half-time, the visitors maybe felt they could have levelled twice, first through Pell who saw his free header inside the area arrow off target.
Then with almost the last kick of the half, Hamilton managed to wriggle in the box but his powerful effort was straight at Covolan.
At the start of the second half with Vale having made a change and switching to a 3-5-2 system, with Ben Garrity replacing Dennis Politic, they came out of the blocks quickly as they had a chance to get a second in the 50th minute.
This was when the substitute slipped Wilson in on the right, but his low shot from a tight angle was fired wide of the near post.
On the hour mark, with Vale slightly gaining control, Savin prevented James Gibbons from extending Vale's lead when captain Tom Conlon flicked the ball into his path, but from the left, the young goalkeeper did well to block the natural full back's powerful strike.
Then with 20 minutes to go, things got even better for the Valiants when forward Bishop was sent off for an off the ball incident involving Nathan Smith making Coleman's task of pegging the game back even harder.
Only two minutes later, the hosts actually could have actually taken immediate advantage of the extra man when on the counter-attack Garrity was found in space from range, but his low shot was excellently tipped round the post by Savin.
It didn't take Clarke's side long to deservedly double their advantage though when Wilson’s clever footwork down the right allowed him space to shoot and his low shot via a deflection trickled past the goalkeeper and into the back of the net.
Despite being a man down, the away side could have halved the deficit when a low cross down the right somehow found its way to Jack Nolan at the back post, but his powerful close-range effort was brilliantly turned behind by Covolan.
With 13 minutes remaining, Vale were so close to adding to their tally when Amoo flicked the ball to Conlon in space, but his low shot was kept out with his feet by Savin.
Only moments later, this is exactly what happened, as it ended up being a memorable day for the former Manchester United striker and this one was the easiest of the lot.
Conlon's cross from the left was sliced by Matt Nottingham into the path of the 25-year-old, who from point-blank range had the simple task of guiding the ball home, confirming that the match ball would be going to him.
With five minutes to go Stanley did grab a consolation when Hamilton's low strike from range, following a short free-kick, was fired into the bottom left-hand corner.
Almost immediately after the goal, Vale restored their three-goal lead with Newcastle United loanee Cass claiming his first goal for the club.
Gibbons once again got down the left and squared the ball to Cass on the far side and his first-time strike was smashed into the back of the net, before in stoppage time substitute Lloyd completed the humiliation after converting a rebound after his original shot was kept out.
Switch of system at the break the difference
Despite being a goal ahead at the break, Vale had allowed the visitors a lot of the ball and seemed happy to just sit back and hit Coleman's side on the counter-attack.
This is obviously something that Clarke wanted to change shown by his half-time substitution which saw the Valiants go from a 3-4-3 system to a 3-52, with Garrity coming into the three in midfield and Wilson and Amoo filling in the two striker positions.
After doing this, Vale started to press a lot more and began to obtain more possession of the ball resulting in the hosts putting Stanley on the back foot.
In the second period, the away side failed to really have a sniff as Covolan hardly was needed to be involved.
This was as the front two always put the defence under pressure, along with the three in midfield winning their battles high up the pitch, allowing the ball to be switched in dangerous areas out wide.
Their constant press and high intensity in that second half was key to Vale achieving such a convincing victory and is typical of the type of football that Clarke has been getting his team to play so far this season.
Star Player- James Wilson
It was a perfect day for the striker after he claimed a hat-trick and it can't be argued that it was his potent finishing that was vital to Vale progressing to the next round of the FA Cup.
His composure when receiving the ball, along with his dangerous runs in behind and the bravery to go at defenders caused Stanley's back three problems all afternoon.
Before today the forward had only scored three goals with strikes against Harrogate Town, Leyton Orient and Colchester United, but his overall play and the way in which he creates plenty of space out wide has provided plenty of encouragement for Vale fans.
However, after a flurry of goals today, he will hope that he can use it as a platform to build on going forward as he will look to add more goals in his game for the remainder of the season