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 saw Pajtim Kasami fire wide from a tight angle.
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.
The second half started as the first finished, with Liverpool on top. First Suarez was denied by Stekelenburg and then Gerrard bent a shot that the goalkeeper did well to keep out.
Suarez soon made amends for his miss, however, with his second of the game. Gerrard stole the ball and played it to the striker, who took a touch before finding the back of the net.
Suarez almost completed his hat-trick when Philippe Senderos's clearance rebounded off the striker and forced Stekelenburg to save.