Nuno Espirito Santo's side then went seven clear on aggregate with a third goal on the night as Ruben Vinagre added to Pyunik's woes.
Diogo Jota celebrates adding Wolves' fourth late on
Image credit: Getty Images
Second-half substitute Diogo Jota helped match the second-leg score by superbly netting a fourth with an overhead kick to see Wolves run out 4-0 winners.
Espirito Santo's men will face a much stiffer test in the final qualifying round, with Torino drawing 1-1 at Belarusian side Shakhtyor Soligorsk to complete a 6-1 aggregate success.
Elsewhere, Rangers are also through after Alfredo Morelos scored twice in a 3-1 win over Midtjylland at Ibrox to set up a Europa League play-off round tie with Legia Warsaw.
Steven Gerrard's side were in a commanding position after winning 4-2 in Denmark a week ago, and strikes from Morelos and Sheyi Ojo in the opening period set them on their way.
Morelos then notched his ninth goal of the season already after the interval, and while Evander pulled one back for the Danes, it is Rangers who travel to Poland next week.
It was not such a good night for fellow Scottish Premiership side Aberdeen, who fell at the Europa League third qualifying round for the sixth straight campaign, losing 2-0 at home to Rijeka to complete a 4-0 drubbing on aggregate.
By the time Funso Ojo was sent off, the Dons were already a goal down through Stjepan Loncar's 10th-minute strike, and Antonio Colak then added a second to kill the tie.