The England goalkeeper was the Toffees' star performer with a string of excellent stops as Chelsea missed the chance to go top of the Premier League above Liverpool, with Manchester City able to reclaim top spot with at least a point against Manchester United.
It was a frustrating afternoon for Maurizio Sarri, who has now set a new managerial record for the best unbeaten run from the start of a Premier League career with 12, with Marcos Alonso hitting the post before Alvaro Morata had a goal disallowed for offside as Everton held firm to halt Chelsea's five-match winning run in all competitions.
Everton's winless run in the top flight at Chelsea stretches to 24, but Marco Silva's men have now been beaten just once in six league outings.
The visitors might have been ahead early on had Andre Gomes kept a header down, while Bernard went close on the half-volley.
5 - There were five yellow cards in the opening 45 minutes between Chelsea and Everton, the most in the first half of a single Premier League game this season. Excitable. #CHEEVE— OptaJoe (@OptaJoe) November 11, 2018
Chelsea crafted their first opening in the 40th minute – Pickford pulling off an instinctive save to keep out Alonso's wicked volley.
Pickford was called into action again after the restart, diving to his right to deny Morata.
Everton responded well, Theo Walcott's poor touch letting Chelsea off the hook before Gylfi Sigurdsson forced Kepa Arrizabalaga into a fine save.
Pickford was at his best again soon after, making two fantastic stops from a pair of Eden Hazard attempts, although Everton's goalkeeper was beaten when Alonso's strike clipped the post.
Everton's resolve seemed to have been broken when Morata tucked home in the 72nd minute, but the offside flag came to their rescue, with Ross Barkley's late introduction against his former club unable to inspire Chelsea to victory.
What does it mean? Chelsea slip up
Chelsea have always seemed to find a way to win so far under Sarri, but despite dominating possession and creating plenty of opportunities in the second half, the Blues' attempts proved fruitless, and with such congestion at the top, a win on Sunday could well have been crucial.
Pickford to the rescue
Marco Silva's team limited the space for Chelsea well, but when the Blues did get through Everton's ranks, they found England goalkeeper Pickford in top form.
Jorginho marked out
Everton set up to be compact, with Sigurdsson crucial in eliminating the space for Jorginho – who was perhaps fortunate to escape with just a booking for a high lunge on the Iceland international – to operate, forcing Chelsea into a more direct approach.
Key Opta Facts:
- Chelsea have remained unbeaten after their opening 12 games to a season in the top-flight for only the second time, also doing so in 2014-15 when they went on to win the Premier League.
- Chelsea boss Maurizio Sarri has become the first manager in Premier League history to remain unbeaten in his first 12 games in charge in the competition (W8 D4 L0), overtaking Frank Clark's previous record of 11 games.
- Chelsea and Everton have now played out eight goalless draws in the Premier League; only Everton versus Liverpool (nine) has seen more in the competition's history.
- Everton are now winless in their last 24 Premier League visits to Chelsea (D11 L13), their outright second-longest run without a win away at any opponent in their league history after a 36-game streak at Leeds United between April 1953 and December 2001.
- This was Chelsea 21st Premier League clean sheet against Everton; against no other opponent have they managed more in the competition (also 21 versus Newcastle United and Tottenham Hotspur).
Chelsea travel to Wembley to face fellow high-fliers Tottenham on November 24, with Everton welcoming Cardiff City to Goodison Park on the same day.