The Toffees were held to a drab draw at Craven Cottage but despite securing a point, they propped up the table after West Ham beat Tottenham.
Having crashed out of the Carling Cup with a midweek defeat at Brentford, they must now wait at least another week to claim their first Premier League win of the season.
Moyes will be the more satisfied manager after Everton controlled long spells of a dire match that did not show any sign of life until the final half hour.
Fulham keeper Mark Schwarzer made a fine save from Yakubu in the 78th minute and the later stages were lit up by a duel between the rivals that the Australian clearly won.
But the afternoon reflected poorly on both teams with each suffering from their lack of firepower having lost key strikers to the treatment room.
Everton made the brighter start but it took a well-timed tackle from their defender Phil Jagielka to deny Clint Dempsey a great chance.
Carlos Salcido sped into the box and squared to Dempsey, who was about to shoot from the edge of the area only for Jagielka to intervene
Tim Cahill, back having missed the last two games with a knee injury, became tangled with Brede Hangeland as Everton threatened and then Pienaar saw an attempt saved by Schwarzer.
Seamus Coleman was proving influential for the Toffees down the right side as Fulham continued to find themselves on the back foot.
Schwarzer had to react smartly to gather a dangerous cross from Phil Neville with Leighton Baines lurking in the six-yard box.
A dynamic run from Baines kept Everton on the front foot and Fulham's problems grew when Stephen Kelly limped off to be replaced by Chris Baird.
Zoltan Gera opened up his box of tricks to skip past Baines before producing a tame shot that failed to trouble Howard.
Better was to come from Fulham when Simon Davies found Salcido, who expertly teed up Dickson Etuhu only for the Nigerian's header to find the arms of Howard.
Mikel Arteta, restored to the Everton line-up after missing his side's Carling Cup exit in midweek, tried his luck from long range but the Spaniard's shot drifted narrowly wide of the right post.
The impressive Salcido stopped Cahill in his tracks with a textbook challenge as the second half got off to an attritional start.
Gera sent a shot across the face of goal and was then replaced by Eddie Johnson as manager Mark Hughes looked to inject some like into his side.
Fulham responded by producing a slick move that saw Dempsey force a desperate save from Howard before Duff's follow up was deflected to safety.
Everton should have crept ahead when Coleman showed athleticism to find the near post where Yakubu failed to connect while under pressure from Schwarzer.
Dempsey then went close as the match finally began to open up.
A brilliant fingertip save of Yakubu's low piledriver showed exactly why manager Mark Hughes battled so hard to prevent Schwarzer from joining Arsenal in the summer.
Schwarzer advanced off his line to keep Yakubu at bay once more before Kamara, playing his first game for Fulham since January, saw a weak penalty shout rejected by Howard Webb.
Yakubu had two chances to snatch victory but his first effort bobbled agonisingly wide of the right post before he missed a sitter with the last kick of the game.