Liverpool were held to a goalless draw at home in their last 16 second leg by modest Portuguese side Braga, who have never even won a domestic league title, and went out 1-0 on aggregate.
Ajax, knocked out of the Champions League earlier this season, were outclassed 3-0 in Russia by Spartak Moscow for a 4-0 aggregate defeat in Europe's second string competition.
Manchester City beat Dynamo Kiev 1-0 despite having Mario Balotelli sent off in the first half but still lost 2-1 on aggregate while a 1-0 home defeat for Rangers against PSV Eindhoven ended British interest.
Braga were one of three Portuguese sides to make the last eight, with Benfica and Porto also going through, while Twente Enschede and Villarreal completed the quarter-final lineup.
Liverpool made a bright start at Anfield but faltered against a strong Braga defence, failing to create many chances. They are now in danger of missing out altogether on European football next season.
In Moscow Alex Meschini, a Libertadores Cup winner with Internacional in 2006, inspired Spartak, setting up the first goal for Dmitri Kombarov and curling in a freekick for the third after half-time. Fellow Brazilian Welliton got the other goal.
Porto, already 1-0 up against CSKA Moscow from the away leg, raced to a 2-0 lead with goals from Brazilian Hulk and Colombian Fredy Guarin before Zoran Tosic pulled one back before the half hour. But they held on comfortably for a 3-1 aggregate win.
Benfica went through with a 1-1 draw at Paris Saint-Germain giving them a 3-2 aggregate win. Argentine Nicolas Gaitan put the Portuguese side ahead while Mathieu Bodmer volleyed the equaliser for the French team before halftime.
Manchester City manager Roberto Mancini was left wondering what might have happened if the notoriously volatile Balotelli had not lost his head.
The Italian, who also missed a header and blocked a shot by team mate David Silva, was shown a straight red card in the 36th minute for a reckless challenge on Goran Popov with the score still goalless.
Against all the odds, City hauled themselves back into the tie when Aleksandr Kolarov fired past unsighted Kiev keeper Oleksandr Shovkovskiy. But they could not muster a second goal despite creating more chances after halftime.
"I am disappointed with the result because we deserved to score at least another goal," said Manchester City manager Roberto Mancini.
"I hope (Mario is frustrated) because I think that with 11 players we would have won this game and gone to the quarter-final."
Zenit, UEFA Cup winners in 2008, looked capable of wiping out their 3-0 first leg deficit against Dutch champions Twente as Roman Shirokov and Aleksandr Kerzhakov put them 2-0 by half-time.
But the second half was
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.