Juergen Klopp's side were 4-0 up within 14 minutes at Anfield thanks to goals by Darwin Nunez, Bobby Clark, Mohamed Salah and Cody Gakpo.
Serbian striker Veljko Birmancevic pulled one back for the visitors before the break, but Liverpool continued to dominate.
Midfielder Dominik Szoboszlai scored the hosts' fifth goal in the 48th minute and Gakpo netted his second of the night to complete Liverpool's rout.
Salah's goal was his 20th of this campaign and he accomplished the feat for the seventh-successive season, becoming the first Liverpool player in their 132-year history to do so.
At the Bay Arena, Patrik Schick scored twice in stoppage time to send Bayer Leverkusen through in the most dramatic of fashions after they trailed Azerbaijan's 10-man Qarabag 2-0 and faced their first defeat of the season.
Xabi Alonso's side came close to losing in Baku last week but for a late leveller for 2-2 there and this time again fought back to win 3-2 on the night and 5-4 on aggregate.
The clear Bundesliga leaders, who are targeting a treble of trophies this term as they are also into the semi-finals of the German Cup, were helped by a red card to Elvin Cafarguliyev in the 63rd minute.
Atalanta advanced 3-2 on aggregate against Sporting after winning the second leg 2-1, while Brighton won 1-0 thanks to a 37th-minute goal from Danny Welbeck against Roma at the Amex Stadium, but exited the competition after suffering a 4-1 aggregate defeat.
Earlier, AC Milan eased into the quarter-finals with a 3-1 victory at 10-man Slavia Prague to secure a 7-3 aggregate win while West Ham crushed German side Freiburg in a record triumph.
Slavia's hopes of swiftly closing the gap from the 4-2 loss in the first leg were shattered after 20 minutes when captain Tomas Holes received a red card for a stamp on Milan's Davide Calabria.
Christian Pulisic, Ruben Loftus-Cheek and Rafael Leao scored in the first half to put seven-times European champions Milan in complete control of the tie.
The second half was uneventful until six minutes from time when Matej Jurasek claimed a consolation goal for Slavia.
West Ham overwhelmed Freiburg 5-0 thanks to goals from Lucas Paqueta, Jarrod Bowen, Aaron Cresswell and a brace from Mohammed Kudus to claim their biggest win in a major European competition and send them into the last eight 5-1 on aggregate.
David Moyes, in his 250th match as Hammers boss, commended his team.
"We looked like scoring tonight and looked the bigger threat. Obviously, we've got two or three exceptional individual players who helped us get goals. It was a brilliant performance for us to win by five," Moyes told UEFA.com.
Benfica clinched their quarter-final spot with a 1-0 victory over Scottish side Rangers courtesy of Rafa Silva's strike in the 66th minute which secured a 3-2 aggregate win.
Despite Villarreal's 3-1 home win over Olympique de Marseille, featuring goals from Etienne Capoue, Alexander Sorloth and Yerson Mosquera, the French side secured a 5-3 aggregate victory thanks to a stoppage-time strike from Jonathan Clauss.
In the Conference league, Aston Villa progressed with a 4-0 win over Ajax after their first leg ended goalless in Amsterdam.
Ollie Watkins (25th), Leon Bailey (60th), Jhon Duran (75th) and Moussa Diaby (81st) were on target at Villa Park.
Villa join Fenerbahce, Fiorentina, PAOK, Olympiacos, Club Brugge, LOSC and Viktoria Plzen in the quarters.
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