Honours even after dull Goodison encounter
The point lifted Chelsea one place to seventh in the table and Everton also moved up one position to 11th.
Everton had the better chances in the first half but Chelsea goalkeeper Mark Bosnich proved equal to efforts by Tomasz Radzinski, Steve Watson and Thomas Gravesen.
Chelsea's Gianfranco Zola should have scored soon after the interval and the Italian also forced Everton goalkeeper Steve Simonsen to tip his free kick onto the bar.
Chelsea suffered a blow when Bosnich, who has only just regained fitness, was forced off with a worrying-looking thigh injury but his replacement Carlo Cudicini made a brave block to deny Everton a late winner.