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.
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
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