Fulham extended their unbeaten run in the Premier League to four matches but had to be content with just a point following another inspired display by Everton goalkeeper Jordan Pickford.
England's number one pulled off a string of fine saves to earn the visitors a fourth clean sheet of the season.
Fulham's top scorer Aleksandar Mitrovic had 10 efforts on goal but through a combination of good goalkeeping and errant finishing was unable to add to his tally of nine goals for the campaign.
The Serbia striker was perhaps lucky to see out the 90 minutes after escaping with just a yellow card following a strong first-half challenge on Idrissa Gueye.
"It was a red card for Mitrovic," argued Everton manager Frank Lampard. "I have nothing against him but he knew, he put his arm up straight away. VAR is there to deal with these things."
Willian was twice denied by Pickford on what was his 265th Premier League appearance, the most for any Brazilian in the English top flight.
And the Cottagers had two unlikely penalty shouts turned down either side of the break, while Everton's Dominic Calvert-Lewin had an effort ruled out for a clear offside.
The point moved last season's Championship winners into seventh place, while Everton are 12th.