One of the most eventful games of Nuno Espirito Santo's tenure was settled by Moutinho's second-half scorcher, with the Gunners having both David Luiz and Bernd Leno shown red cards.
Mikel Arteta's men saw a Bukayo Saka strike disallowed before they deservedly took the lead through Nicolas Pepe.
Wolves' confidence appeared to be at an all-time low, but they were soon handed a big lifeline. Luiz was sent off as he gave away a penalty on the stroke of half-time, which Ruben Neves coolly dispatched.
Moutinho then came up with what will be looked back upon as one of the best goals of the season before Leno was also sent for an early shower, for handling the ball outside his box.
Finally, Wolves have returned to winning ways in the Premier League. A rollercoaster 90 minutes.
One of the strangest football matches you are likely to ever see. It had everything and, most importantly, Wolves finally got their first league victory of 2021.
In the first half, if looked bleak. Arsenal were toying with Nuno's anxious players and a thrashing seemed to be in the offing, such was the shocking defending on show.
But things miraculously turned around. It was one unbelievable moment after the other.
Firstly, Luiz's dismissal came out of nowhere, with Neves coming up with a composed finish from 12 yards to level the scores.
Moutinho's majestic strike was out of this world, breath-taking – words cannot really do it justice, in truth.
And then Leno shocked everyone with a clear handball, as Arsenal finished two men down.
So, Wolves' midfielders stepped up at long last. The area has come in for criticism as of late, and rightly so, but Neves and Moutinho showed their class when needed.
What a relief these three points are, and what a night this was.
Nuno's team selection – Neves and Adama Traore both returning to the starting XI – sparked debate among Wolves supporters, wondering whether they would be lining up with three or four at the back.
Also noteworthy was that they named eight substitutes instead of the permitted nine, with Leander Dendoncker and Ki-Jana Hoever those dropping to the bench.
Arsenal, meanwhile, welcomed Saka back from injury while star striker Pierre-Emerick Aubameyang made the squad after missing their previous three games for personal reasons.
The mass confusion over Wolves' set-up was cleared up at kick-off as Max Kilman, a centre-half by trade, was on the left of a defensive four. A strange decision from Nuno, Kilman's task was to keep Saka quiet.
But the Gunners man gave him the runaround right from the first whistle.
In the opening minute, Saka burst past him to latch onto a lofted pass and crashed a close-range shot off the woodwork.
The England winger then forced Rui Patricio into a low save before appearing to open the scoring – not being tracked and finding the far corner from near the penalty spot.
Thankfully for Wolves, Alexandre Lacazette was found to be offside in the build-up by the VAR. An enormous let-off, a wake-up call.
Nuno's lot, though, remained fast asleep for ages. Kilman – trying his best but clearly not a left-back – was being targeted time and again by Arsenal.
He entered the book for a late challenge on Saka, and Moutinho was also cautioned for hacking down Lacazette.
You could just sense the sheer panic they were playing with. It was only a matter of time until the visitors got themselves a goal.
And it came with the help of some shambolic defending from Wolves. Pepe easily powered his way past Nelson Semedo and curled the ball home with his weaker right foot.
Soon after that, Pepe – having seen another strike sharply tipped onto the crossbar by Patricio earlier in the half – was just about denied on the counter by Neves.
To say Wolves were at sixes and sevens in defence is putting it lightly. They were all over the shop.
However, in a scarcely-believable, dramatic turn of events, they went into the break level – and with a man advantage.
Daniel Podence played Willian Jose through on goal, and the Brazilian was tripped by compatriot Luiz for a penalty. He was sent off and Neves calmly tucked away the spot-kick.
A get out of jail free card for Wolves. It was absolutely astonishing, and things soon became even more mind-blowing.
The Gunners, who brought on defender Gabriel for striker Lacazette, could do nothing to stop Moutinho's other-worldly effort from finding the back of the net.
Lifting his head up, from 30 yards out, the veteran launched a heat-seeking missile off the inside of the post and past Leno, at full-stretch.
It was a truly remarkable, special goal – Moutinho's first of the campaign – and with it, Wolves' fear dissipated.
Looking a completely different side to the one they were in the first half, they were operating with swagger. Pedro Neto went on a mazy run and may have been about to add a third, but Gabriel slid in to deny him at the last second.
Jose also stung the palms of Leno before a complete moment of madness from the German keeper. Another crazy incident.
Semedo played a ball over the top aiming for Traore and Leno, bizarrely, raced outside his area and batted the ball away with his right arm.
Especially with VAR, there was no hiding how blatant the handball was and he was also dismissed, putting Arsenal down to nine men.
Wolves pressed for another goal and should have got one as Jose failed to connect with Traore's cross but, ultimately, they saw it through. A huge victory for Nuno's charges.
Wolves (4-3-3): Patricio; Semedo, Coady, Boly, Kilman; Neves (Dendoncker, 74), Moutinho, Podence (Vitinha, 62); Traore, Jose (Silva, 90), Neto
Subs not used: Ruddy (gk), Hoever, Richards, Otasowie, Gibbs-White
Goals: Neves (pen, 45+4), Moutinho (49)
Arsenal (4-2-3-1): Leno; Bellerin, Luiz, Holding, Cedric; Xhaka, Partey (Runarsson, 75); Pepe (Aubameyang, 61), Smith Rowe, Saka; Lacazette (Gabriel, 46)
Subs not used: Chambers, Elneny, Ceballos, Odegaard, Willian, Martinelli
Goal: Pepe (33)
Red cards: Luiz (45+3), Leno (73)
Referee: Craig Pawson (Sheffield)