Aleksandar Mitrovic had the best of the home side's chances, but the Serbia striker was unable to add to his nine-goal Premier League haul for the season.
Jordan Pickford made a number of key saves to keep Fulham at bay, including from Mitrovic and Willian, as the Cottagers were left frustrated after failing to break the deadlock despite having 24 attempts on goal.
Fulham went into the game in seventh place and having scored 22 goals this season - before the weekend only Arsenal, Manchester City and Tottenham had found the net more times - but they drew a rare blank against manager Marco Silva's former club.
The game started with both sides trying to gain an early advantage, Willian heading wide for Fulham while Bernd Leno had to be alert at the other end to tip a Demarai Gray header over the bar.
Pickford made two saves in a matter of minutes to keep put Willian's shot and then a well-timed header from the home side's leading scorer Mitrovic.
There were chances from both sides as they continued to search for the opener, Tim Ream forcing a save from Pickford before Leno had to gather a header from James Tarkowski.
Fulham had a penalty shout at the start of the second half when Willian went down after an Idrissa Gueye challenge in the area, but referee John Brooks waved play on and there was no lengthy VAR review.
Everton worked hard to break down Fulham's attempts to open them up while committing limited numbers forward themselves, relying mainly on playing counter-attacking football and finding it difficult to escape Fulham's offside trap.
Mitrovic had another opportunity in the 61st minute when he was picked out by Bobby Decordova-Reid at the far post. The prolific forward controlled the ball on his chest before taking a shot but the ball flew harmlessly over the crossbar.
Fulham continued to pile on the pressure but carving openings was difficult, with the Everton defenders making a number of blocks and Pickford proving an unbeatable final line of defence, including keeping out a low drive from Willian in the closing stages.