Ryan Leak's first goal for the Brewers sent the home side into a deserved 1-0 lead at the break.
However, after only coming on the pitch with 16 minutes remaining, two quickfire goals by the Bolton Wanderers loanee ensured that the comeback was complete to round off a memorable day for the club.
Story of the game
It was the Valiants that came out the brighter with them pressurizing the hosts from the off, but it was Burton that had the first shot of the tie inside seven minutes, coming off the boot of Kane Hemmings from range, but it was easily held by Aidan Stone.
However, with Jimmy Floyd Hasselbaink's outfit having settled into the game they broke the deadlock on the 23rd minute mark.
This was when Hemmings' inviting cross from the right was latched on to by Leake at the back post who powered his header past Stone and into the bottom left-hand corner.
The opener seemed to give the Brewers a new lease of energy as only two minutes later Jonny Smith got down the right and forced a stop out of the back-up goalkeeper at his near post.
Then moments later, Smith was inches away from doubling his side's advantage if it wasn't for an excellent block from Dan Jones.
The winger was found in space down the right again and fired a low shot towards the goal that beat Stone, but Jones did brilliantly to recover and deflect the shot on to the bar and out to safety.
The home side were really finding their feet now and were fortunate to not be more ahead this time through Hemmings whose effort from close range just failed to be diverted on target.
Against the run of play, the visitors forced Ben Garratt into action for the first time with six minutes until the interval.
This was when following good flowing play, James Gibbons was slipped down the right, his cutback found David Amoo who smashed his effort goalwards, but it was turned behind by the Burton captain.
A minute before the break with no doubt Burton having had the better first 45, they were close to extending their advantage when a Deji Oshilaja wayward shot found Ben Hamer on the left but his curling shot bent off target.
However, with almost the last kick of the first half Vale were adamant they should have had a penalty when a Gibbons cross was met by Garrity who got to the ball first and went down inside the area, but the referee waved away the claims sparking outrage from the visiting fans.
After the break, it was Vale that started with more intent and should have been level on 47 minutes when Tom Pett played through Amoo who clipped the ball past Garratt but the ball dribbled narrowly wide of the right-hand post.
Vale, now playing in full confidence, saw another opportunity go begging two minutes later when a Gibbons cross was cleared as far as Pett on the edge of the box but his smashed shot sailed well over the bar.
Inside 63 minutes, Daniel Jebbison went on a solo run but couldn't find the clinical touch as his final finish was wayward.
With the game seeming to run its course, Vale's comeback began with 11 minutes to go as two second-half substitutes combined to send the game level.
David Worral was found on the right and cut the ball back to Politic and he fired a deflected shot into the top left-hand corner.
Then only three minutes later, Clarke's side turned the tie on its head when Devante Rodney threaded Politic through on goal and he had the composure to round Garrat and smash the ball home.
The hosts did have one final chance to send the game to a replay in stoppage time, but from point-blank range, Shaughnessy didn't put enough on the ball as Stone held on, seeing Vale progress through to the third round of the FA Cup.
Strength of character to comeback
With 15 minutes to go, even sending the game back to Vale Park in an FA Cup replay looked unlikely especially due to the injury crisis the squad have in the striking roles as ahead of the game they only had one fit natural striker in the form of Rodney.
So, testament must be given to Clarke and his team for fighting through it in what has to be said were difficult conditions and carving out the opportunities to ensure that they secured themselves a spot in the hat for the third round draw.
Despite not having many options in terms of natural forwards, credit for the comeback has to be given to Clarke's substitution with of course the most notable one being Politic, who came on and got the decisive goals that saw Vale come away with their first-ever win at the Pirelli Stadium.
It wasn't only the loanee that made a difference though, Worrall who has been a threat going forward all season for the club, came on and looked dangerous as he often whipped dangerous balls into the area that caused problems for Burton's back three.
It can't be forgotten that the winger was also the one to grab the assist that started off the comeback, with it being his clever ball into Politic that allowed him to smack the ball home.
Star Player- Dennis Politic
Even though the 21-year old was only on the pitch for around 20 minutes, his double is ultimately is what saw Vale into the next round of the competition.
His quick and sharp movement along with his will to drive forward with the ball really caused problems for the home side's defence and provided Clarke with a different option off the bench.
This is considering that throughout the game, Amoo and Rodney were struggling to really find themselves in space in dangerous positions.
Politic's brace at the Pirelli Stadium means that in 11 appearances he has accumulated five goals which is a feat considering that he has only started six games in all competitions during his time so far at the club.