Liverpool scored four times inside 14 minutes in a 6-1 thrashing of Sparta Prague to reach the quarter-finals of the Europa League as Bundesliga leaders Bayer Leverkusen needed two stoppage-time goals to eliminate Qarabag.
AC Milan, Marseille, Roma and West Ham were among the other sides to also book their place in Friday's quarter-final draw.
Despite leading 5-1 from the first leg, Jurgen Klopp named a surprisingly strong Liverpool side and was well-rewarded as the Reds remain on course for a quadruple of trophies.
Darwin Nunez and 19-year-old Bobby Clark got Klopp's men off to a flying start before Mohamed Salah marked his first club start since January 1 with his 20th of the season.
Cody Gakpo fired in the fourth before Liverpool briefly took their foot off the gas ahead of Sunday's trip to Manchester United in the FA Cup.
Veljko Birmancevic pulled a goal back for Sparta, but Dominik Szoboszlai and Gakpo completed the rout in the second half.
"It was an amazing win – and an amazing first 15 minutes," said Gakpo after an 11-2 aggregate victory.
"It's what we had in mind. It shows the confidence in the team at the moment."
Leverkusen still unbeaten
Leverkusen boss Xabi Alonso has been heavily linked as Klopp's successor at Anfield, and his side remain unbeaten in all competitions this season thanks to Patrik Schick's stoppage-time double to beat Qarabag 3-2 on the night and 5-4 on aggregate.
Schick also struck in added time during the first leg as Leverkusen came from 2-0 down and they needed to do the same at home.
Abdellah Zoubir and Juninho gave the Azeri champions a stunning lead.
But the visitors paid for having to play the final half hour down to 10 men as Elvin Cafarguliyev was sent off for hauling down Jeremie Frimpong.
Frimpong reduced the arrears before Schick turned home at the near post on 93 minutes.
And there was still time for the Czech forward to head in the winner seven minutes into added time.
West Ham trailed 1-0 from the first leg against Freiburg but roared back to reach a European quarter-final for the third consecutive season.
Lucas Paqueta turned home a corner at the far post before Jarrod Bowen blasted home to turn the tie around before half-time.
Aaron Cresswell arrowed a shot into the far corner and Mohammed Kudus then took centre stage to secure a record European win for David Moyes' men.
The Ghanaian ran from deep inside his own half through the heart of the Freiburg midfield and defence to slot in the fourth.
And Kudus then smashed home from the edge of the box.
"Obviously we've got two or three exceptional individual players who helped us get goals," said Moyes. "It was a brilliant performance for us to win by five."
Brighton left themselves too much to do as a 1-0 win over Roma could not overturn a 4-0 first-leg deficit.
Danny Welbeck scored the only goal as the Seagulls at least ended their first-ever European campaign on a high.
Benfica ended Rangers' hopes of another deep run in the Europa League with a 1-0 win in the lashing Glasgow rain to progress 3-2 on aggregate.
Rafa Silva produced a composed finish from Angel Di Maria's through ball 24 minutes from time and the goal stood after initially being ruled offside.
Seven-time European champions AC Milan cruised into the last eight with a 3-1 win over 10-man Slavia Prague.
Three quick goals from Christian Pulisic, Ruben Loftus-Cheek and Rafael Leao killed off the tie before half-time.
Villarreal ended Jean-Louis Gasset's unbeaten run as Marseille boss but a 3-1 victory in Spain was too little, too late for the Spaniards as they went out 5-3 on aggregate.
Atalanta beat Sporting Lisbon 2-1 on the night in Italy to go through 3-2 on aggregate.
AFP