The Portuguese side will host the first leg on April 1 against the Premier League team, whose only hope of silverware this season now lies with Europe's secondary club competition.
"We know they are a good team and of course we have played them before in the Champions League," Liverpool coach Rafa Benitez told Liverpool's website.
"They are a massive club with a lot of support from their fans and it will be a tough game.
"They are top of the table and playing really well at the moment. I know a number of their players well - Aimar, Saviola, Garcia - so we know they have a lot of quality."
Fulham, who dumped Juventus out of the competition with a memorable 4-1 win on Thursday, were rewarded with a tie against German champions VfL Wolfsburg, who beat Rubin Kazan with a goal in the last minute of extra-time.
Unfashionable Fulham are making their first appearance in a European quarter-final.
Valencia and Atletico Madrid will meet in an all-Spanish tie while Hamburg SV, who beat Anderlecht 6-5 on aggregate, face another Belgian side in Standard Liege.
Draws for the Europa League quarter-finals and semi-finals made on Friday.
QUARTER-FINALS
Fulham v VfL Wolfsburg
Hamburg SV v Standard Liege
Valencia v Atletico Madrid
Benfica v Liverpool
SEMI-FINALS
Valencia/Atletico Madrid v Benfica/Liverpool
Hamburg SV/Standard Liege v Fulham/VfL Wolfsburg
The quarter-final first legs will be played on April 1 with the second legs scheduled for April 8.
The semi-final first legs will be played on April 22 with the second legs scheduled for April 29.
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