The Blues were guilty of squandering several chances against the Toffees, but the visitors were well worth their point from Sunday's fixture in West London.
Maurizio Sarri's side remain unbeaten in the top flight this season but have now recorded four draws from seven top-flight fixtures, while Marco Silva's Everton improve their recent return in the Premier League to 13 points from six games.
Everton went into the contest having failed to register a league win at Stamford Bridge since 1994, but the Toffees started brightly in West London and Andre Gomes could have done better with a header from a corner.
Chelsea were struggling to get into the game, and Everton almost went in front just before the quarter-of-an-hour mark when Bernard fizzed a half-volley from 25 yards just wide of the post.
Marcos Alonso curled a free kick just off target, but chances were few and far between at a time when tensions were beginning to boil over due to a harsh yellow card handed to Yerry Mina for a challenge on Eden Hazard.
Chelsea had under-performed during the opening 45 minutes but they almost went ahead after just 15 seconds of the second period, with Alvaro Morata seeing a volley kept out by Pickford after he had met a cross from Hazard.
Everton responded well with Gylfi Sigurdsson seeing a curler from the edge of the area tipped over the crossbar by Kepa Arrizabalaga, while Bernard also failed to make the most of a chance at the back post.
However, the home side were soon back on the front foot and Willian should have done better after being played through on goal, only to drag a low strike marginally wide of the far post.
Hazard also had a long-range effort tipped behind by Pickford while Alonso - a constant threat down the left flank - saw a volley clip the outside of the far post before going wide of the target.
Everton remained a threat on the break but it was all Chelsea, although another opportunity was squandered with 17 minutes left when Hazard poked wide from 16 yards after being sent clear of the Toffees backline.
The chances dried up during the closing quarter-of-an-hour as Everton moved to five at the back, but the visitors will be delighted with a rare point from a city where they have now only won once in 17 attempts.
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Rudiger, Luiz, Alonso; Kante, Jorginho (Fabregas 64'), Kovacic (Barkley 82'); Willian (Pedro 68'), Morata, Hazard
EVERTON (4-2-3-1): Pickford; Coleman, Keane, Mina, Digne; Gueye, Gomes; Walcott, Sigurdsson (Jagielka 76'), Bernard (Lookman 64'); Richarlison (Calvert-Lewin 89')