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 defense 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, which responded in fine style 24 hours after fierce rival Manchester United moved to the brink 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 defense 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 15 yards (meters).