The Reds made a quick start and Daniel Agger had the first shot of the match after only two minutes, but his volley was too high and flew over the crossbar.
Fulham's first foray into the Liverpool area resulted in Pajtim Kasami taking down a long ball and firing wide from a tight angle.
However, Martin Jol's side soon fell behind. Luis Suarez flicked a ball towards the back post, but Fernando Amorebieta got the last touch as the ball went in.
It got worse for the away side three minutes later as Liverpool doubled their advantage. Gerrard delivered into the box and Martin Skrtel lost his marker to head into the bottom corner.
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Phillipe Coutinho then forced Stekelenburg into a save as Liverpool pushed for a third. However, that goal did come as Suarez beat the goalkeeper with a finish from a tight angle after Jordan Henderson had slipped the ball in.