Trailing 3-2 after being beaten at home, the Danes eked out a 2-0 victory in Madrid to win 4-3 overall. Morton Bisgaard's clincher, following a 71st minute strike from Ulrik Pedersen, came in the 91st minute.
"Digesting this loss is like digesting poison," Real Madrid coach Jorge Valdano said after the game.
The Spanish club is a six-time winner of Europe's most prestigious prize, the Champions Cup. Odense has never even made the final of any of Europe's club competitions but now has made the last eight of the UEFA Cup.
On a bad night for Spain, Athletic de Bilbao and Deportivo de La Coruna, also tumbled out leaving no Spanish clubs in contention. Yet three Italian clubs moved smoothly into the final eight.
Italy's Parma, runner up last season in the Cup Winners Cup, overturned a first leg deficit to beat Bilbao 4-2, Juventus, a two-time UEFA Cup titlist in the past five years, downed Admira Wacker 2-1 and Lazio gained a 2-1 victory over Turkey's Trabzonspor.
Gianfranco Zola scored Parma's first in the 21st minute. Dino Baggio made it 2-0 by halftime and added a third soon after the break.
Oscar Vales replied for the Spaniards but Parma's Portuguese midfielder, Fernando Couto, netted a fourth for the Italians. Julen Guerrero scored another for Bilbao but Parma held on.
Already 3-1 from the first leg in Austria, Juventus moved further ahead through defender Ciro Ferrara in the 17th minute before substitute Gerd Wimmer replied for Wacker 16 minutes from the end. GianLuca Vialli made sure for Juventus four minutes from time.
Lazio, leading 2-1 from the away leg in Trabzon, moved further ahead after 25 minutes through a strike by defender Roberto Cravero. Soner Boz tied the game 17 minutes from the end, but substitute striker Marco Di Vaio netted Lazio's winner two minutes later.
Two German clubs also made it to the final eight.
Borussia Dortmund overcame Deportivo 3-1 after overtime after trailing 1-0 from the first leg. Michael Zorc fired the Germans ahead five minutes into the second half and the two teams were tied at 1-1 overall after 180 minutes.
Alfredo scored for the Spaniards in extra time but Karl-Heinz Riedle and Lars Rickel fired two more for Dortmund for a 4-2 overall triumph.
Bayer Leverkusen romped into the last eight with a 4-0 crushing of Poland's Katowice, giving it an 8-1 aggregate advantage. All four goals came in the opening 28 minutes.
Veteran midfielder Bernd Schuster scored the first and Andreas Thom, Heiko Scholz and Pavel Hapel added the others as the Poles, already 4-1 down from the home leg, were swept aside.
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