Premier League - Saha injury mars Everton win
That made it 2-0 with man-of-the-match Seamus Coleman having opened the scoring with a 36th minute header but Fulham fought back and Clint Dempsey produced a fine effort in the 62nd minute.
The Cottagers pressed for an equaliser with a purpose that was missing before they went 2-0 down but Everton hung on for a win that pushes them level on points with seventh placed Bolton, with only goals scored keeping the Trotters above them.
However, they will be concerned about Saha who looked in agony after a falling badly on his ankle after going up for a header.
Fulham are 12th but only three points above the relegation zone, and they have now lost 18 in a row at Goodison - an astonishing record that goes back to 1959.
Fulham were toothless in the opening 50 minutes which allowed the Toffees to cruise to a 2-0 lead.
The hosts nearly scored in the 28th minute when Saha's header looked set for the corner of the net but Carlos Salcido headed off the line.
Soon after the restart and it was two as Saha smashed a free-kick into the corner of the net, although he was only allowed to do that as Danny Murphy broke off the wall leaving a gap that the ball fizzed through.
However, Fulham finally awoke from their slumber in the final 40 minutes, helped by the introduction of Bobby Zamora who gave them a focal point in attack.
His lovely check and roll in one sweeping movement saw him guide a Damien Duff cross into the path of Dempsey and the American then placed an excellent low drive from the edge of the area into the corner of the net for his career-best 11th league goal of the season.
Suddenly it was Fulham who were dominating possession although they didn't create a clear chance until five minutes from time when Zamora uncharacteristically fluffed a header when another substitute, Gael Kakuta, had picked him out with a fine ball.
Just before that, Salcido headed off the line again, this time from Phil Jagielka, but the key late moment for Everton could be Saha's injury in the 88th minute.
They got the three points but will miss the Frenchman's goals if he is out for any extended period of time.
What is more Tim Cahill also had to depart earlier in the match with a foot injury, although his knock did at least look less serious.