Andraz Sporar put the Slovakian outfit in front early on as the visitors were second best in the first half.
However, Wolves turned on the style in the second period and won thanks to a long-range strike from Romain Saiss and a Raul Jimenez penalty despite Diogo Jota's dismissal for two yellow cards.
The result sees them leap-frog Slovan into second in Group K, with three of the six group stage games played.
This was far from your normal football match. Fortunately, Wolves are a pretty special team.
The atmosphere – almost a full house of under-14s jumping up and down, waving their arms, and loudly shouting 'Slovan' throughout – was surreal, writes Joe Edwards in Bratislava.
It was not Galatasaray's 'Welcome to Hell', of course. It was the polar opposite – Tehelne pole transformed into a children's playground – but, to be fair, it would have been enough to unsettle most sides.
Nuno Espirito Santo's bunch were not themselves in the first half either. But they fought through for a fantastic victory.
Upon hearing about Uefa's behind-closed-doors sanction against the hosts, it would have been fair to expect the ground to be empty.
But the under-14s – allowed in for free, with an adult accompanying every 10 – packed it out, with 20,333 the official attendance, and certainly made themselves heard.
Wolves, meanwhile, were controversially only able to sell 200 tickets, with many making the trip to Slovakia's capital city and not being able to go to the game.
Those that did, thankfully, were treated to a tremendous comeback.
Adama Traore came on at half-time, after Andraz Sporar had given Slovan the advantage with a clever finish, and Wolves were so much better after that.
Jimenez began to see more of the ball, while the midfield – despite having one less man in there with a change to a 3-4-3 formation – gained more control.
Saiss levelled the scoring as an ambitious attempt had the desired effect, while the relentless Jimenez was taken down for a penalty which he calmly put away.
This was not a free-flowing display or an example of completely outclassing the opposition.
Slovan, to be fair to them, were a well-drilled outfit with some impressive players going forward.
Nuno's lot, even with Jota's late sending-off, just had too much character and pulled off such a satisfying success.
Youngster Maximilian Kilman was handed a huge opportunity on the left side of defence, with the 22-year-old making only his third senior start.
Real Madrid loanee Jesus Vallejo was left out of the squad entirely following his poor display as a substitute in last weekend's 1-1 draw with Southampton.
Saiss, who had been struggling with a hamstring problem, started in midfield alongside Joao Moutinho, with teenager Morgan Gibbs-White in a more advanced role.
Another teen, Taylor Perry, was among the substitutes, as was Jota, who is still working his way back to full fitness after a toe injury.
The build-up to the encounter was quite something, with the children in the stands emphatically taking part in a series of chants led by the stadium announcer.
The orchestration over the PA system continued after kick-off too, with a microphone seemingly being passed around for the youngsters to scream whatever they wanted.
On the pitch, Wolves sprayed the ball around calmly amid the chaos in the first few minutes. But come the 11th, Slovan were in front.
Sporar beat the offside trap and latched onto an over-the-top pass before catching Rui Patricio cold as he found the net.
The striker, who scored 29 league goals for the Slovak Super Liga champions last season, took the shot early and wrong-footed the Euro 2016-winner, with the ball flying in at the near post.
And Slovan were the better side for the rest of the first half, with Moha constantly asking questions of Matt Doherty, and Sporar finding space with clever movement. David Holman had a good chance from the edge of the area after Jimenez was caught napping, too.
Wolves, meanwhile, had Gibbs-White miss the target by a few feet with a long-range effort while Willy Boly went close with a header – rising highest to meet Moutinho's free-kick.
Ruben Vinagre also had the odd promising foray forward, but the hosts were playing at a higher tempo and deserved the lead they had entering the interval.
A poor pitch, as well as the crazy atmosphere, was difficult for Wolves to cope with. A lot more was definitely needed, though.
Nuno recognised that too as he made a change for the start of the second period, bringing on Traore for Patrick Cutrone, who continuously lost possession in the first 45, and switching to 3-4-3 – Gibbs-White going to the left of the front three.
That switch immediately brought a great opening for Wolves. Traore played the ball into Doherty, and he found Jimenez, who turned and fed Gibbs-White.
From seven yards, though, the 19-year-old could only shoot straight into Dominik Greif's grateful hands.
Still, the benefits of the tactical shake-up were there for all to see. Nuno's charges were playing with greater confidence and purpose.
The goals came to put Wolves in front, as well.
Firstly, Saiss, with a challenge not forthcoming from a white shirt while 25 yards out, threw caution to the wind – and it paid dividends.
The Moroccan struck the ball sweetly with his left foot and, while Greif got a hand to it, it found the far corner.
Five minutes later, Wolves scored again. Jimenez was pushed over as he tried to get on the end of a cross, and a penalty was awarded.
Unsurprisingly, the £32million man took the spot-kick and coolly converted it, sending Greif the wrong way in the process.
Right after that second goal, Jota was introduced – and he ended up being sent for an early shower.
Two unnecessary bookings within seconds each other saw Wolves play the final few minutes plus added-on time with 10 men.
To Nuno's delight and the 200 gold and black supporters in the ground, they held on for a terrific triumph.
11 - Goal. Slovan take the lead through Sporar. Against the run of play, he gets in behind the Wolves defence and surprises Patricio, taking the shot early and wrong-footing the keeper.
36 - Not far away from an equaliser. Moutinho whips in an inviting free-kick and Boly meets it well, but the header ends up sailing narrowly wide for Nuno's side.
48 - Big chance for Wolves! Doherty finds Jimenez with a low cross from the right. The Mexican turns and plays it to Gibbs-White, who shoots straight at Greif from seven yards.
58 - GOOOOOAAALLLLLLLL! Wolves are level! Saiss tries his luck from range and finds the net with a powerful left-footed strike. Grief got a hand to it but was not able to keep it out.
63 - GOOOOOOOAALLLLLLLLL!! Nuno's men, who have been so much better in this second half, are ahead! Jimenez is pushed over in the box for a penalty before he sends the keeper the wrong way from 12 yards.
87 – RED CARD! Jota's off. The substitute has only just been booked, and then he lunges in for another. Silly. Wolves down to 10 men for the last few minutes plus stoppage-time.
Slovan (4-2-3-1): Greif; Medvedev, Abena, Bozhikov, De Marco; Ibrahim (Cavric, 81), De Kamps; Daniel (Ratao, 66), Holman (Ljubicic, 57), Moha; Sporar
Subs not used: Sulla (gk), Drazic, Nono, Bajric
Goal: Sporar (11)
Wolves (3-5-2): Patricio; Boly, Coady (c), Kilman; Doherty, Moutinho, Saiss (Dendoncker, 75), Gibbs-White (Jota, 59), Vinagre; Jimenez, Cutrone (Traore, 46)
Subs not used: Ruddy (gk), Otto, Neves, Perry
Goals: Saiss (58), Jimenez (63)
Red card: Jota (87)
Referee: Yevhenii Aranovskyi (Ukraine)
Wolves return to Premier League action at Newcastle on Sunday (2pm).