Pedro Neto scored on his debut as Wolves brushed Pyunik aside 4-0 at Molineux to secure an 8-0 aggregate victory and a place in the Europa League play-off round.
Nuno Espirito Santo's men were already virtually guaranteed of progression but ensured they moved two games away from the group stage in style, although Serie A side Torino are next up in the competition.
Neto's opening goal for the club came early in the second half and he was quickly involved again, setting up Morgan Gibbs-White.
Ruben Vinagre added Wolves' third and they completed the scoring in stunning fashion thanks to an acrobatic finish from Diogo Jota.
Wolves had established a dominant position in the tie thanks to last week's 4-0 win at the Republican Stadium, where Raul Jimenez scored twice.
Having handed debuts to Jesus Vallejo and Neto in the return fixture, they almost took an early lead when the latter fired wide from outside the box, while Patrick Cutrone wasted two chances before half-time.
Andrija Dragojevic stayed big to deny Gibbs-White after the restart but Neto struck the opener from close range in the 54th minute, tucking in a Cutrone cross.
Confidence was then flowing for the 19-year-old as he darted into space on the left and put the ball on a plate for the arriving Gibbs-White to net his first senior goal.
Pyunik were cut open on the right flank again in the 64th minute when Adama Traore set up Vinagre for his second Europa League strike of the season.
The pick of the goals was still to come with three minutes left, however, a fine team move seeing Wolves find Jimenez in space and his scooped pass over the Pyunik defence was met by Jota's wonderful overhead kick.
Torino will likely provide a far sterner test, but Wolves continue to enjoy their first taste of European football since the 1980-81 campaign.