Blues were reduced to NINE men by the 65th minute due to second-half sendings-off, Josh Williams then Josh Andrews seeing red.
Williams went two minutes after Andrews had equalised at the start of the second-half.
And when the big frontman Andrews received his marching orders, Paul Robinson's side were totally up against it.
But Blues organisation, their game-management and durability, added to a sure threat on the break, repelled everything that Barnsley could throw at them.
The telling blow came as the tie seemed destined for extra-time.
Hurst flicked the ball over an opponent, darted on into the area behind the defensive line and cracked a half-volley that went in off the post.
It was a brilliant goal and it sparked scenes of unbridled joy in the Blues ranks.
But Blues still had to negotiate the remaining minutes, plus five added on.
They did so, the backline marshalled by captain Nico Gordon with schoolboy Rico Browne alongside him, and could even have added a third, when dazzling footwork by Adan George in a tight spot near the byline set up Hurst for another shot at goal which slammed into the sidenetting.
When the final whistle went Blues bench erupted and the substitutes, staff - Robinson did a Mourinho - and non-playing members of the squad streamed onto the pitch and mobbed those out there in celebration.
In contrast, half-a-dozen Barnsley players slumped to the deck, face down or head in hands, trying to make sense of what had happened.
On a difficult, bumpy pitch at Leamington FC, Blues had their problems at first as Barnsley started strongly, working furiously to close them down and interchange passes quickly to put the defence under pressure.
Aiden Marsh wriggled free inside the box and had a great chance to score but stuck his left foot shot wide of the post.
After the initial, frenzied exchanges settled down somewhat, Blues began to impose themselves and create opportunities
Blues sprang from deep in the 17th minute, moving the ball into George whose backheel flick invited Andrews to let fly, but his powerful attempt cleared the crossbar.
Barnsley went into the lead from a corner that Josh Bradley-Hurst couldn't quite grasp tightly enough, the ball escaping to be headed in from a few yards by Rudi Pache (27 minutes.)
Blues came back when Andrews turned in midfield and pinged a lovely crossfield pass to Williams, who travelled forward and played the ball into the heart of the area but unfortunately George couldn't get a sweet enough contact to convert.
Just before the break again Blues spread play wide, via George, and the dangerous Williams accelerated into the area and blasted high from a tight angle.
Barnsley - who lost Pache to injury shortly after his goal and had to readjust, dropping midfielder Jordan Helliwell in at centre-half - defended tigerishly whenever Blues came at them with their movement and pace.
A fast start to the second-half by Blues reaped reward in the 49th minute when Andrews broke through two opponents on the right of the area and thumped an emphatic low shot diagonally just inside the post.
But two minutes later Williams caught Sam Nicholson's shins in a tackle as they both went to ground competing for the ball; the referee came over with a yellow card in his hand ready to administer, but after speaking to his assistant instead brandished red.
Williams had been causing Barnsley problems and it meant George was moved out to the right and Andrews left as the focal point up top.
Blues did well to thwart Barnsley's attacks as they went forward at them with renewed vigour.
Then, in the 65th minute, as Andrews tried to get away from Helliwell on halfway during a counter attack, the Barnsley player stuck to him tightly and appeared to catch him with an elbow. As they went to ground Andrews kicked out and the referee raced over to the incident to produce a red card again.
With so long left, you would have thought Blues were doomed - but not so.
Bradley-Hurst made a smart reaction stop with his feet at the near post when Barnsley did get in on the right.
But that was the only real testing save he had to make to the finish as Blues got their shape right and blocked up the passages down the middle.
Remi Walker - then Amari Miller -Tate Campbell and Hurst formed a trio in front of the back four and George now adopted the central striker's role.
They did a great screening job and when Barnsley were forced wide, the excellent pair of Lucas Powell and Marcel Oakley at full-back never let the odds get the better of them.
As Barnsley grew ever more frustrated as Blues defended their box and their lines magnificently and also tried to be poised when possible, Hurst provided that stunning moment.
He had threatened to get away a couple of times as Blues soaked it up and went forward on the break when the chance arose.
And his goal had an element of magic about it, the way he conjured himself into the area with nothing else on and then produced such a devastating finish.
The victory, for nine-man Blues, will doubtless go down in FA Youth Cup folklore - and not surprisingly. An incredible night, indeed.
Blues: Bradley-Hurst, Oakley, Powell, Campbell, Gordon, Browne, Walker (Miller 70), Hurst (Djrovic 90+4), Williams, Andrews, George (Traore 90). Subs not used: Clayton, Zohore, Kinina, Finnegan.