Zoltan Gera scored on both sides of the break before Fulham's United States midfielder Dempsey grabbed the fourth to send the unfancied Premier League side into the last eight.
Juventus were reduced to nine-men in added time when defender Jonathan Zebina saw red for kicking out at Fulham's Ireland winger Damien Duff.
"It was a fantastic team performance. Their goalkeeper made four outstanding saves," Fulham manager Roy Hodgson said.
"Beating Juve is something special. I'm sure every one of them [Fulham fans] will remember this night for some time to come."
In another drama-filled match, prolific World Cup-bound Spain striker David Villa scored a hat-trick for Valencia as they drew 4-4 at Werder Bremen to go through on away goals.
Benfica, held 1-1 at home by Olympique Marseille in the first leg, were in more trouble when the French side's Senegal forward Mamadou Niang volleyed home from close range in the 70th minute.
But the Portuguese team created more chances and deservedly levelled four minutes later when Argentine Maxi Pereira exploited a blunder by the Marseille defence to score with a low drive from just outside the box.
Confusion reigned after Brazilian Alan Kardec scored the winner for Benfica with an angled shot inside the area from a Pablo Aimar free kick.
Kardec was yellow-carded for taking off his shirt in celebration and Marseille winger Hatem Ben Arfa, who had only just come on, was then sent off for kicking Javi Garcia.
In the other games, Dieudonne Mbokani gave Standard Liege a 1-0 win over Panathinaikos as the Belgian side completed a 4-1 aggregate victory over the Greeks.
Liverpool overcame a 1-0 first leg defeat to go through 3-1 on aggregate at Anfield thanks to a pair of sublime dinks from Torres and an early penalty from Gerrard.