Mason Mount gave the hosts a deserved lead in the 70th minute with a calm finish past Everton goalkeeper Jordan Pickford.
But the Toffees equalised just four minutes later through 19-year-old Jarrad Branthwaite, who converted at the back post after Anthony Gordon's free-kick.
Thiago Silva had a chance to win it for Chelsea but was denied by Jordan Pickford.
The England goalkeeper was superb throughout the contest, which lifts Everton eight points above the drop zone.
It was another poor result for Chelsea, who are now four points behind league leaders Manchester City.
Thomas Tuchel's side now have one win in their last four matches.
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