It wasn't a bad point for the Cottagers, who extended their run in the league to four games, but had to be content with a point.
A string of fine saves by England's number one earned them a fourth clean sheet of the season.
After two efforts were turned away, the top scorer in the league was unable to add to his 9 goals.
Willian is the most experienced Brazilian in the English top flight with 265 appearances.
The Cottagers were denied two unlikely penalty shouts either side of the break, while Dominic Calvert-Lewin had an effort ruled out for being in an ineligible position.
The point moved last season's Championship winners into seventh place, whileEverton are back up to 12th after their victory at Crystal Palace.
Going into this weekend's games, only three teams had scored more goals than the Cottagers in the league.
Marco Silva says the current form is a continuation of the attacking philosophy put in place as they rattled in 106 goals last season.
Pickford was England's top defenceman and he kept the free- scoring Cottagers at bay.
He produced three saves in quick succession in a breathless start to the game, tipping over Willian's shot on the turn and a long-range effort from Harrison Reed.
It was hoped that returning to the side after an injury would boost their tally of just 11 goals in 13 games.
Tim Ream's freehead from the centre bounced just wide of Pickford's far post as the Cottagers tried to break through the Toffees' back-guard.
The referee checked his monitor and ruled out any claims for a penalty after James Tarkowski tangled with Bobby Decordova-Reid.
The VAR once again ruled out a penalty shout for Silva's side after Willian went down following a stretched challenge from Gueye.
As the game wore on, the pace dropped and the Cottagers had to settle for a draw.
The goalless stalemate was a perfect example of all that has been good and bad in the early stages of the season.
The only time they have scored more than one goal in a game this season was in their previous match, when they beat Palace by three goals.
They looked to have gotten a boost in the opening 10 minutes from the return of Calvert-Lewin.
While Pickford was called to action at the other end,Everton had their own chances to open the scoring, with Anthony Gordon's close-range shot blocked by Robinson and Leno tipping over Demarai Gray's stinging drive from the edge of the box.
Gray set up a chance for Calvert-Lewin, who got enough on his low cross to divert the ball away from Gordon, who was diving to his right.
The visitors tried to wrestle the initiative away from the Cottagers before the break.
Lampard's side looked to be more than happy to take the point and move on to the next match.
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