Cardozo pens sink 10-man Liverpool
The other three first-leg ties also produced plenty of twists and turns on another exciting night in a competition often derided as the poor relation of European club football.
Ruud van Nistelrooy's winner gave Hamburg SV a 2-1 success at home to Standard Liege, the Germans also coming from behind, while David Villa scored a late equaliser to help Valencia draw 2-2 at home to Atletico Madrid in an all-Spanish clash.
Fulham - who thumped Juventus 4-1 at home in the last round to reach their first European quarter-final - continued their fine run with a 2-1 victory over German champions VfL Wolfsburg whose late goal took the gloss off the home side's performance.
"We have got to be happy that the European adventure is very much alive here at Craven Cottage and we have taken another very famous scalp along the way," said Fulham manager Roy Hodgson.
The match was billed as one of the west London club's biggest but Liverpool felt very differently about their trip to the Estadio da Luz to face Benfica in a clash of two former European champions.
Rafael Benitez's team, in real danger of missing out on qualifying for next year's Champions League, are looking on the Europa League as a mere consolation in a miserable season.
Liverpool were given a great start when Daniel Agger's skilful flick silenced the home crowd after nine minutes.
But their evening began to unravel after half an hour when their Netherlands forward Ryan Babel was sent off for a shove in the face of Benfica defender Luisao.
The hosts, starting with seven South Americans, equalised in the 59th minute when Cardozo buried a harshly-awarded penalty after Emiliano Insua tangled with Pablo Aimar.
Cardozo produced a repeat 20 minutes later, again finding the bottom right corner of Pepe Reina's goal after Jamie Carragher had blocked an Angel di Maria cross with his arm.
Benitez told reporters: "Always when decisions are against a team you don´t like it but because you cannot change them now the only way is to try be positive and... think about our fans behind the team and to play with all the players at Anfield."
Several firecrackers were thrown onto the pitch and the referee threatened to stop play after one had left a large hole in the turf near the touchline.
Hamburg, whose stadium will host the final, fell behind when Democratic Republic of Congo striker Dieudonne Mbokani was left free at a corner and scored with a powerful header in the 31st.
Ex-Manchester United and Real Madrid striker Van Nistelrooy hit the post and then shot wide from 12 metres before Croatia forward Mladen Petric levelled with a 42nd minute penalty.
Three minutes later, Van Nistelrooy got in front of his marker to guide Dennis Aogo's cross home from six metres.