Rodriguez made it seven goals in the last three league matches by scoring twice in the first six minutes, his first coming after just 32 seconds. The Argentina midfielder added the team's fourth with a stunning long-range shot in the 70th.
Netherlands forward Dirk Kuyt had already put the Reds 3-0 ahead in the 16th with his fifth goal in successive games and Uruguay striker Luis Suarez completed the rout in the 75th, capping a sensational individual performance that ran Fulham's defence ragged and putting the Reds two points clear of Tottenham in fifth place.
Moussa Dembele and Steve Sidwell scored second-half consolation goals for the hosts but they were outclassed by Liverpool, who responded in fine style 24 hours after fierce rivals Manchester United moved to the brink of surpassing the Reds to clinch a record 19th English title.
An irresistible opening 16 minutes of the match saw Liverpool tear the hosts apart, scoring three times and threatening many more. Suarez, playing on his own up front in the continued absence of the injured Andy Carroll, didn't get on the scoresheet in that period but it was his trickery and nonstop running which created massive gaps in Fulham's defence that his teammates exploited to the full.
Suarez was the instigator of the first goal, collecting a pass from Brazil midfielder Lucas Leiva, cutting into the box but seeing his cross deflected off Carlos Salcido. Fulham goalkeeper Mark Schwarzer bizarrely attempted to kick the ball away instead of picking it up and Rodriguez pounced on the rash clearance by smashing home a finish low into the net from 15m.
Worse was to follow five minutes later for Fulham when fullback Glen Johnson ran into acres of space down the right flank and stood up a deep cross that Rodriguez volleyed in left-footed at the far post.
Brede Hangeland was fortunate to escape a red card when he appeared to haul down Suarez in the area in the 15th, with referee Lee Mason referring to his linesman before ruling play on.
But Fulham's glaring defensive deficiencies were shown up again seconds later, Kuyt given an unforgivable amount of time and space at the edge of the area to collect a Johnson pass, turn and send in a low shot that Schwarzer parried into his own net at the near post.
Rodriguez and Suarez missed more decent chances as it continued to be one-way traffic, with the home side's only response coming when United States midfielder Clint Dempsey had a shot cleared off the line by Johnson.
The introduction at halftime of substitute Bobby Zamora gave Fulham more presence in attack after the break and it was the England striker who set up Dembele's goal, laying the ball off for the Belgium international to curl in from 12m.
But Rodriguez snuffed out any chance of a comeback, easing away from the leaden-footed Hangeland and smashing a stunning effort from 25m into the top corner for his 10th of the season.
Suarez got the goal his selfless performance deserved when he ran on to substitute Jonjo Shelvey's through-ball to round Schwarzer and tap in before Sidwell fired in a half-volley in the 86th to complete the scoring.
Jamie Carragher capped a successful night for Liverpool by making his 666th club appearance, taking the ex-England centre back above Ray Clemence and Emlyn Hughes into second place on the all-time list.
