In a dramatic second leg in Ukraine, the Spaniards won 3-2 with an extra-time goal from Uruguayan Ernesto Chevanton which put them through 5-4 on aggregate in the second knockout round tie.
Sevilla trailed 2-1 and were on the brink of elimination until, in a last throw of the dice, Palop ventured forward in the fourth minute of stoppage time and headed in a Daniel Alves corner to force an extra 30 minutes.
"Of course it was a miracle that we equalised in the last seconds," coach Juande Ramos told a news conference.
"But we only have ourselves to blame for not putting this game away earlier."
The holders were joined in the last eight by AZ Alkmaar, who overturned a 4-2 first-leg loss to defeat Newcastle United 2-0 at the DSB stadium and progress on away goals.
Espanyol, the losing 1988 finalists, became the third Spanish side into the last eight after ending Maccabi Haifa's run. Espanyol romped to a 4-0 home win to go through by the same score on aggregate.
Former European champions Benfica complete the quarter-final line-up, winger Simao Sabrosa netting a late penalty in Lisbon to seal a 3-1 win (4-3 aggregate) and knock out Paris St Germain.
England's Tottenham Hotspur, German sides Werder Bremen and Bayer Leverkusen and Spain's Osasuna reached the last eight on Wednesday.
Shakhtar had appeared set for the quarter-finals when Brazilian Elano, a second-half substitute, fired the hosts into a 2-1 lead seven minutes from time.
An enthralling match also featured an audacious "scorpion-kick" goal from Shakhtar's Matuzalem. The Brazilian met Olexiy Bielikj's cross with an outrageous back-heeled flick, reminiscent of Colombian goalkeeper Rene Higuita's infamous scorpion kick save against England in 1995.
Newcastle chose the wrong time to demonstrate their defensive frailties, allowing AZ to extend their unbeaten home streak in Europe to 29 matches.
Louis van Gaal's side needed at least two goals without conceding one themselves and got them through Shota Arveladze and Danny Koevermans.
Georgian Arveladze turned home a low right-wing cross from Koevermans after 14 minutes and the provider turned scorer 11 minutes into the second half, powering home a header from a corner.
Espanyol strikers Walter Pandiani (nine) and Luis Garcia (six) took their tally in this season's competition to 15 with a goal apiece as the Primera Liga side ruthlessly crushed Maccabi Haifa 4-0.
The Catalans scored three times in the space of nine minutes, early in the second half, with goals from playmaker Ivan de la Pena and strikers Raul Tamudo and Garcia. Pandiani added a fourth at the end.
There was late drama in Lisbon where Benfica made their relentless pressure tell against PSG, winning a late penalty after midfielder Youssuf Mulumbu had scythed down Leo.
Mulumbu was sent off and Simao beat Mickael Landreau from the spot.
Trailing 2-1 from the first leg, Benfica took a 2-0 lead through Simao and Petit before Portuguese forward Pauleta hauled the French side level on aggregate.
The draw for the quarter-finals takes place on Friday at Hampden Park, the venue for the final on May 16.