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Liverpool will now travel to Torres' former club Atletico Madrid on April 22 and host the second leg the following Thursday.
"We played against a good team and knew we'd have to work hard," Liverpool manager Rafa Benitez said. "We took advantage of their weaknesses and the third goal was really, really good. Fernando will be excited about going back to his old club."
Liverpool crashed out of the Champions League, the competition it won in 2005, at the group stage and the Europa League now offers the only chance of ending a four-year trophy drought.
"We've beaten a very good side, a top team, and it's a very good result," Liverpool defender Jamie Carragher said. "We can take massive confidence from this as they are flying in their league and next season will probably be in the Champions League."
Unlike Liverpool, which is sixth in the English Premier League, four points adrift of the fourth Champions League spot, Benfica was top of its domestic league and on a 27-game unbeaten run before Thursday.
Benfica pressed from the start but Liverpool gradually gained a foothold and Dirk Kuyt headed the home side in front in the 28th minute after latching onto Steven Gerrard's corner.
The goal was initially ruled out as the referee's assistant on the right flank flagged for offside. But after Torres led Liverpool's protests, referee Bjorn Kuipers reversed the decision and Kuyt was able to end a 1,084-minute goal drought.
"There was a bit of confusion, but it was a corner so it couldn't be offside," Kuyt said. "The linesman thought someone on our team touched the ball on so it was offside but he changed his mind and we had the goal. The rules are the same in every country. It wasn't offside, it was a goal."
There was no disputing Lucas Leiva's strike after he completed a swift counterattack six minutes later.
Having being released by Yossi Benayoun, Gerrard sent a defense-splitting pass through to the advancing Lucas, who rounded goalkeeper Julio Cesar and slotted into the net.
"We had a very good period in the first few minutes but Liverpool is a very experienced team," Benfica coach Jorge Jesus said. "We couldn't cope with the speed of Liverpool's counterattacks, which were very fast."
Another lightning-quick break forward saw Liverpool go 4-2 in front on aggregate in the 59th minute. After successfully defending a Benfica free kick, Javier Mascherano dispatched the ball to Benayoun, who raced down the left flank before passing to Kuyt on the right.
The Netherlands forward crossed to Torres, who fired the ball into the net from six meters (yards).
Liverpool looked to be cruising into the last four, but Oscar Cardoza poked a low free kick through the defensive wall to make it 4-3 with 20 minutes to go.
Nerves heightened at Anfield when Cardoza took another free kick shortly afterward, but his shot was deflected wide and Torres netted again in the 82nd.
Mascherano sent Torres through on goal and the Spain striker clipped the ball over Jose Moreira, who had just replaced the injured Julio Cesar.
"He has a little injury but he kept going and scored two fantastic goals for us," Kuyt said of Torres.