Jelavic - who joined from Rangers in late January - scored twice and hit the post, while Marouane Fellaini and Tim Cahill also netted, as the hosts were helped by some comical Whites defending.
The victory marks three consecutive Premier League matches in which Everton have scored four, boding well for next season if they manage to overcome their usual poor start.
The result also means Fulham miss the chance to move above the Toffees into seventh, with David Moyes’s side now five points clear of the Whites and Liverpool, but too far short of Chelsea to make a run for sixth.
In the absence of manager Martin Jol, who missed the game through illness, Fulham - now without a win at Goodison in 11 Premier League matches and 19 league games overall - appeared to write the match off, defending like schoolchildren as Everton netted with all three first-half shots on target.
Those goals also came from dreadful errors: a needless sixth-minute handball from Pavel Pogrebnyak gave Jelavic his first from the spot, while two shocking attempts to clear from Damien Duff and Aaron Hughes let Fellaini and Jelavic again find the net.
Fellaini’s second goal in two matches came on 16 minutes, just after Croatia forward Jelavic had hit the woodwork, the Belgian’s header allowed inside the far post by a terrible miskick from Duff, who would have cleared simply by standing still.
Jelavic’s second came five minutes before the break, the ex-Rangers man initially forced wide by Mark Schwarzer but somehow allowed to finish inside the near post as both the Australian goalkeeper and Aaron Hughes failed to clear, the latter aping Duff with a wild swing when standing his ground would have sufficed.
Fulham, meanwhile, went close through the usual source that is Clint Dempsey, who had one deflected effort tipped on to the post by a brilliant Tim Howard save, the American goalkeeper also denying Kerim Frei, who was lively for the visitors.
But the match was long lost, Fulham conceding a fourth - although this time it was a fine goal as a scooped pass from Steven Pienaar released Cahill, who had started the move and completed it with a low finish, his second goal in an injury-hit season.
Everton had chances to add further to their lead, Schwarzer making good stops from Pienaar and Leon Osman, who had another excellent match as he racked up his 300th appearance for the Toffees.