Antonio Conte's Premier League champions had the better chances, with Willian's effort cleared off the line in a first-half scramble.
Marcos Alonso was also denied by a last-ditch clearance from substitute Ashley Williams just after the break, before the Everton defender deflected a Victor Moses cross on to his own bar.
Everton soaked up the visitors' pressure and nearly claimed three points when Michael Keane headed over in stoppage time.
Everton are six matches unbeaten and remain ninth. Chelsea stay third.
Everton: 26 Dec v West Brom (A)
Chelsea: 26 Dec v Brighton (H)