First-half strikes courtesy of Danny Murphy and David Elm set the ball rolling and Bobby Zamora rounded things off after the break.
While Burnley suffered their 12th defeat away from home, Fulham prepared for some mouth-watering challenges ahead by collecting an easy three points.
Roy Hodgson's men face Notts County in the fifth round of the FA Cup on Sunday before a Europa League date with destiny against Shakhtar Donetsk, but things are looking bleak for Burnley.
An 18-year-old Brian Laws scored the opener in Burnley's 2-0 victory against Fulham in 1980 and the Clarets have not tasted victory at Craven Cottage since.
Now manager of the club, Laws finally got off the mark last weekend with a 2-1 victory at home to West Ham, but his side continue to toil with a travel sickness that has seen them pick up just point away from Turf Moor.
They started well enough, with Andre Bikey looking lively, but Fulham packed more of a punch going forward and Simon Davies came agonisingly close to meeting Damien Duff's cross after eight minutes.
Such is the fragility of the Burnley defence that it was only a matter of time until Hodgson's men took the lead.
Nicky Shorey's punt forward was nodded into Murphy's path and the former Liverpool star bundled the ball past Brian Jensen to give his side a 22nd minute lead.
Dickson Etuhu almost doubled the lead six minutes later but his stinging drive was acrobatically pushed to safety by Jensen.
The Burnley goalkeeper, known as 'The Beast', then made a ghastly error 15 minutes before the interval to gift Fulham a crucial second.
Zamora's shot took a slight deflection but Jensen could only spill it at the feet of Elm, who tucked home from three yards for his first goal for the club.
Zamora should have inflicted more misery on the visitors five minutes after the break but his low shot was palmed round the post by Jensen.
He didn't have to wait long to add his name to the scoresheet, however. The Fulham striker, with whom Hodgson reckons has an outside chance of an England call-up, curled in a sublime free-kick on 52 minutes to make it three.
The hardy souls that travelled from Burnley remained in good voice and Laws will have to count on their support as the season reaches a critical stage.