Former Chelsea striker Sturridge cancelled out Dimitar Berbatov's opener in the first half before going on to complete his treble with two strikes in an impressive second-half showing.
Philippe Coutinho had the first noteworthy effort of the game on 15 minutes when he beat two Fulham challenges on the edge of the box and fired in a swerving shot which Mark Schwarzer did well to bat away.
After a lacklustre opening half hour Fulham broke the deadlock after 33 minutes with their first serious effort on goal.Sascha Riether sent over a fine cross from the right to the unmarked Berbatov to head past Pepe Reina from close-range to score his first goal since the start of April.
The lead proved short-lived as Liverpool equalised three minutes later with Sturridge latching onto a long ball upfield from Andre Wisdom before turning Aaron Hughes inside out and smashing his right-footed shot high into the net past Schwarzer.
Fulham had strong claims for a penalty turned down just past the hour mark when Bryan Ruiz's cross appeared to strike the arm of Lucas, but Halsey waved away the home side's appeals.
Liverpool grabbed the lead on 62 minutes when Coutinho's deflected shot fell into the path of Sturridge inside the six-yard box and the striker gleefully slotted the ball into the net past Schwarzer.
Sturridge missed a great chance of complete his hat-trick a minute later when he robbed Brede Hangeland of possession to go clear on goal, but Schwarzer did well to deny Sturridge.
Five minutes later Sturridge fluffed another great opening to grab a hat-trick when he burst clear on goal and instead of passing to two team-mates unmarked inside the box he fired in a shot which Schwarzer did well to keep out.
Liverpool continued to squander chances to add to their lead as Stewart Downing picked out the unmarked Jonjo Shelvey inside the box, but just when the midfielder was about to pull the trigger he lost his footing allowing Hughes to get back to clear the danger.
Fulham came close to equalising twice 15 minutes from time when Reina tipped Berbatov's shot round the post and from the resulting corner the Liverpool keeper produced a superb flying save to keep out Hughes' goalbound header.
Fabio Borini was inches away from scoring on 78 minutes with almost his first touch after coming on when he cut inside and smashed a shot off the post with Schwarzer well beaten in the Fulham goal.
Sturridge finally completed his hat-trick and put the game beyond Fulham five minutes from time when Coutinho sent him in on goal and he beat Schwarzer to the ball to lift his effort into the net.