After dominating the first half Everton faced a much tougher second 45 minutes with Fulham battling for every ball and crashing two efforts against the woodwork.
David Moyes' men went into the game full of confidence on the back of a 1-1 draw against Manchester United, and a 1-0 win at Bolton in midweek.
Everton piled the pressure on the visitors back line from the off and within the first two minutes a high ball into the box from captain Phil Neville found the head of the free-scoring Marouane Fellaini.
The towering midfielder expertly directed the ball to the on-rushing Tim Cahill but his swing miss allowed Fulham to clear their lines.
Making his first start for Everton in a month, Cahill looked sharp and it was his determination on 10 minutes that won the ball deep in the Fulham half and released Mikel Arteta down the left.
His whipped in cross flew straight to the head of Steven Pienaar but the young midfielder seemed surprised by the ball's pace and could only direct his effort into the arms of a grateful Mark Schwarzer.
Having been camped in their own half for most of the opening 20 minutes, Fulham struggled to play their way around a solid Everton defence but it was Jimmy Bullard who had the first strike at goal for the visitors but his 25 yard-effort only found Row Z.
The visitors began to string a few passes together and a ball onto the chest of Bobby Zamora on the edge of the Everton area was accurately played off to his strike partner Andy Johnson.
The former Everton striker rolled the ball to Bullard who once again drilled from distance but failed to trouble Tim Howard.
Everton looked the better side going forward and should have taken the lead just before the half-hour mark after Arteta had played a nice one-two with Yakubu. The Spaniard drove a low pass across the edge of the box to Fellaini who was looking for his fourth goal in five matches.
The 20-year-old blasted his shot past a sprawling Schwarzer only for the ball to hit the foot of the post and no-one was following up to turn home the rebound.
Johnson had the best chance of the half when Zamora challenged Blues keeper Howard at the front post and.
Under pressure from the former Spurs and West Ham striker, Howard could only palm the ball to the feet of Johnson but, after deciding against a first-time shot, he allowed the Blues to get back and clear the danger.
The second half was a much tighter affair with Fulham chasing down every ball and making the Everton midfield work twice as hard as in the first 45 minutes but it was Moyes' side that had the first chance to open the scoring.
Finding himself on the left with only wing-back John Paintsil to beat, Arteta performed a single step-over to give himself the room to whip a ball in to Cahill who powered a header inches over the bar.
Roy Hodgson had fired up his players at half-time and the response was almost instant when Zamora's cross was met by Zoltan Gera who smashed his header against the bar.
Fulham were again unlucky on 72 minutes when Bullard's 25-yard free-kick was heading towards the top corner only for Howard to somehow get a hand to the ball and palm it down.
It went straight to Zamora who shot across the face of goal only to once again be denied by the woodwork.
Everton thought they had snatched the lead when a long punt from Howard took one bounce and invited a challenge from substitute striker Victor Anichebe on Schwarzer who dropped the ball into his own net.
However, the whistle went for a foul much to the bewilderment of the entire Everton squad.
The game was won with only three minutes left on the clock after a battling Leon Osman fought to keep hold of the ball deep in the Fulham half and his high cross was met by the head of Saha - his first Toffees goal against his former club.