The Croatia international scored twice to take his tally to 10 in 13 matches since arriving from Rangers in January and an impressive nine in his last seven appearances.
With manager Martin Jol absent through illness the Cottagers barely put up a fight as the Toffees made it 12 goals in their last three league matches.
With four against Sunderland and then their remarkable comeback against Manchester United at Old Trafford last weekend, David Moyes' side became the first Everton team since 1964 to score four times in three successive league matches.
Fulham never came close to ending their woeful run of never winning a league match at Goodison Park. It now stands at 19, the longest sequence for a top-flight club at one particular ground.
In the build-up to the game Moyes had spoken of the belief and encouragement Jelavic's goals had given the side, and while three months may not be long enough to provide a full evaluation of long-term success he shows all the signs of being capable of producing on a regular basis.
What he has brought to the team in a short space of time is a cutting edge and end product.
On many occasions this season Everton's build-up play has been good but they have lacked the necessary threat in the penalty area. Not so now with Jelavic, whose first was a straightforward penalty - albeit created by his own free-kick - and his second an impressive finish from the narrowest of angles.
The 26-year-old was up and running as early as the seventh minute when Pavel Pogrebnyak raised his arm in the defensive wall and referee Phil Dowd pointed to the spot without hesitation.
Jelavic sent Mark Schwarzer the wrong way and almost embarrassed the goalkeeper again when he crashed a low shot against the near post after Phil Jagielka's long ball had dropped just beyond Brede Hangeland.
Everton's second was not long in coming as, courtesy of some lax defending, the unmarked Marouane Fellaini was able to head in Steven Pienaar's 16th-minute corner with Damien Duff's flailing leg on the line failing to clear the danger.
In half an hour Fulham had tested Everton goalkeeper Tim Howard just once, when he tipped over Kerim Frei's shot, but the goalkeeper was mightily relieved to see the ball drop behind after he had finger-tipped Clint Dempsey's deflected shot onto the crossbar.
The match was over as a contest five minutes before half-time when Pienaar's through-ball allowed Jelavic to outsprint the defence, draw out Schwarzer and then somehow fire a shot from a narrow angle through the goalkeeper's legs.
Fulham's misery continued after the break as on the hour Everton substitute Tim Cahill made it four after good link play between himself, Leon Osman and Pienaar.
The latter's scooped pass over the defence was met by his midfield colleague's perfectly-timed run to slide a low volley past Schwarzer for only his third of the season.
Jelavic was denied a deserved hat-trick when the Fulham goalkeeper saved his 25-yard free-kick and Pienaar forced another save out of Schwarzer with a shot across goal but any further goals would have merely been window-dressing.
Fulham failed to create any chance of significance in the second half and their performance was a typical end-of-season one for a side in mid-table with nothing to play for.
Prior to the game Moyes had admitted he was happy for the season to end quickly as they too had no real incentive other than finishing above near-neighbours Liverpool. But he must now be wondering what they could have achieved with their current form had there still been a month or more to go.
And had they not had such a poor first half of the campaign, the opportunities could have been even greater.