Neto, one of four players to make their full Wolves bow along with Patrick Cutrone, Jesus Vallejo and Max Kilman, turned in Cutrone cross shortly after half-time.
Neto turned provider to tee up Gibbs-White for his first Wolves strike, before Adama Traore laid one on a plate for Ruben Vinagre as Wolves scored three times in 10 minutes.
They saved the best for last when Diogo Jota sent a stunning overhead kick past the keeper to make it 4-0 in the closing stages as Wolves set up a play-off clash with Italian side Torino next week.
The standard of opposition should be taken into account, undoubtedly.
But, in beating first Crusaders and now Pyunik while scoring a total of 14 goals, Wolves had made an accomplished start to life in the Europa League, writes Tim Spiers at Molineux.
Nuno Espirito Santo's team have been both clinical and professional in equal measure, coping well with the extra games on a Thursday, the travel and their defensive-minded and highly-motivated opponents.
Nuno called for his players to get their shooting boots back on after firing a blank at Leicester on Saturday. Well, they certainly did that during a blistering 10-minute spell shortly after half-time and then Diogo Jota added what will, even on August 15, be one of the moments of the entire season with a sublime overhead kick.
Perhaps the most pleasing aspects of a very positive evening were that Nuno could rest nine players and not see the performance level from last Thursday drop, plus his four full debutants all played their part in a comfortable victory, most notably the gusty Patrick Cutrone who didn't stop running all night and displayed a fantastic attitude and work rate, plus a good technique, as well as the lively and pacey Pedro Neto who grabbed a goal and an assist.
Jesus Vallejo sent through some excellent passes from the back and Maximilian Kilman improved after a shaky start to give a solid display.
All in all then, a very successful opening few weeks to a long campaign for Nuno's team. Onto Turin...
Nuno made nine changes from the team that drew 0-0 at Leicester, with only Conor Coady and Joao Moutinho keeping their places.
Patrick Cutrone and Maximilian Kilman were handed their first Wolves starts, while forward Pedro Neto and centre-half Jesus Vallejo were seen in a Wolves shirt for the first time.
The tie was, of course, all-but over, so for Wolves the night represented an opportunity to see what their fringe players and debutants had to offer ahead of a busy period of five matches in 14 days before the international break.
In front of a near full house at Molineux there were some nervous touches from Kilman and Vallejo in the early stages but, after a sluggish opening, they and Wolves soon settled and tried to prise open the Pyunik defence.
In terms of momentum and build-up play, a fresh-faced team showing so many changes was understandably lacking on both fronts, but Wolves created numerous chances as the half went on.
Cutrone was the stand-out player of the half with a performance of enterprise, resourcefulness and positivity. The Italian popped up all over the place, doing his fair share of defensive work and looking to turn and run at defenders or float in behind for a through ball.
The latter approach almost produced an excellent goal after a superb raking pass from Vallejo, but Cutrone crashed wide from a tight angle on the right after beating his man.
Cutrone also showed ingenuity and wonderful technique when sending an improbable 20-yard volley whistling over the bar on the turn with a superb effort.
Support act Neto, all left-foot and pace, sashayed around the final third with good close control and ability, which almost led to a debut goal when he drilled wide after cutting menacingly in from the right flank.
Ruben Vinagre and Adama Traore also impressed, with the former crisply striking a 25-yard bullet that didn't clear the bar by far, while on the other flank Traore had the beating of his man – sometimes two men – for fun but lacked that killer final ball or a decent shot, as displayed when a left-footed effort went out for a throw.
Cutrone also headed a Joao Moutinho free-kick over the bar and narrowly missed getting on the end of Vinagre's ball across goal.
It was a surreal atmosphere at times with no away supporters frequenting Molineux – understandable given both the scoreline and the cost of travelling to England from a country with far lower living standards – but one enthusiastic flag-waving Pyunik supporter had infiltrated the Billy Wright stand, whose less-than-impressed regulars were threatening to descend from gentle murmuring to collective tutting, akin to a riot in what is generally the quietest part of the stadium.
Wolves' dominance continued after the break – and the Pyunik defence soon wilted as Nuno's team gave the crowd what they wanted.
Three very similar goals were quickly despatched in an impressive 10-minute spell as Wolves gave the aggregate scoreline a healthy look.
First, three of Wolves' new signings assembled as Vallejo clipped a perfect ball forward for Cutrone who got to the byline and cross for Neto who couldn't miss from six yards.
Four minutes later the trick was repeated. This time Neto turned provider from the left, drilling for Gibbs-White who gleefully accepted the simple task of finishing from very close range to score his first Wolves goal in front of the South Bank. Nuno literally jumped for joy and the smile on Gibbs-White's face said it all – the 56-game wait for his debut Wolves goal was finally over.
Wolves were rampant now and after just another six minutes it was 3-0. Yet again the method didn't change but the personnel changed – Traore left Pyunik left-back Armen Manucharyan on his backside kicking at thin air before crossing low for Vinagre, who scored his second goal in two Molineux appearances this season having also netted against Crusaders.
Pyunik were flagging and the last thing they will have wanted to see was Diogo Jota and Raul Jimenez entering the fray, which they did when replacing Neto and Cutrone who were both rightly give rousing receptions.
The game soon petered out but Wolves had one moment left in them. And boy was it worth waiting for.
Jimenez nonchalantly flicked the ball in the air and Jota spectacularly greeted it with a perfect overhead kick from 15 yards out. It was a quite stunning goal that Molineux rose to acclaim – and a fitting finish to a very satisfactory evening.
54 - GOAALLLL! Wolves break the deadlock, and it's Neto with a debut goal. Cutrone latches onto Vallejo's long-ball and squares it across goal, leaving Neto with a simple tap-in. Nice move.
58 - GOOAALLLLL!! A swift second for Nuno's side as Gibbs-White scores for the first time in a competitive senior game. Neto turns provider, drilling it low for the Stafford-born teenager to poke home.
64 - GOOOAALLLLLL!!! Three to the good, with a goal very similar to the first two. Traore gets to the byline and finds Vinagre, who smashes the ball into the roof of the net from a few yards.
88 - GOOOOOAAALLLLL!!! What a stunner from Jota to cap off a great night's work for Wolves. The ball loops up in the air in the box and Jota, so gracefully, finishes on the overhead kick. A magical goal from an exceptional talent.
Wolves (3-5-2): Ruddy; Vallejo, Coady (c), Kilman; Traore, Saiss, Gibbs-White, Moutinho (Dendoncker, 52), Vinagre; Neto (Jimenez, 72), Cutrone (Jota, 72)
Subs not used: Patricio (gk), Bennett, Otto, Neves
Goals: Neto (54), Gibbs-White (58), Vinagre (64), Jota (87).
Pyunik (4-5-1): Dragojevic; Stankov, Zhestokov, Belov, Manucharyan; Efimov, Vardanyan, Mkrtchyan (c), Simonyan (Edigaryan, 71), Shevchuk (Mahmudov, 75); Miranyan
Subs not used: Kobozev (gk), Grigoryan, Marku, Galimov, Burzanovic
Referee: Donatas Rumsas (Lithuania)
Wolves host Manchester United in the Premier League on Monday. They will travel to Turin to face Torino in the first leg of their Europa League play-off next Thursday.