The Toffees were able to frustrate Antonio Conte's men as home stopper Jordan Pickford pulling off a string of smart saves, denying them at every turn and almost snatching all three points towards the end.
It was another game in which Chelsea failed to break down a team who sat back, much to the dismay of the travelling supporters. The champions were able to come away with a point to kick off the festive period, but it was a performance that left a lot to be desired.
Twice off the line
The visitors started brightly, almost going a goal in front in the opening minutes as Victor Moses whipped in a dangerous cross but Eden Hazard was unable to latch onto the end of it. The two best chances of the game came after ten minutes as Phil Jagielka was on hand to clear off the line twice. First Tiemoue Bakayoko was able to get a shot off, but Jagielka was there to clear before the Everton defender blocked the ensuing Willian effort.
Chelsea continued to push in the opening stages, with Hazard heavily involved to no one's surprises. His low cross on 15 minutes was parried away by Pickford before Bakayoko struggled to get his follow up off.
Everton's first glimpse of goal came after half an hour as Tom Davies found space on the edge of the box, but his shot rolled well wide. Minutes after Davies' effort, Chelsea responded with Bakayoko feeding Pedro, but the Spaniard's shot was saved well by Pickford.
Same Old Story
It was a similar story after the break. With over 25 minutes to go, Hazard took his turn to test Pickford and the Belgian's low driven effort forced the 'keeper to get down and push the shot away. Chelsea came inches away from taking a deserved lead with 15 minutes left, but Moses' fizzing cross was deflected onto the crossbar by Ashley Williams.
Everton almost snatched all three points in stoppage time as Gylfi Sigurdsson's corner was met by the head of Michael Keane, but the defender's header flew wide of the post. After Friday's thrilling 3-3 between Arsenal and Liverpool in the Premier League, it was a 0-0 draw to kick things off on Saturday afternoon.
It was a frustrating afternoon for Chelsea, who will feel like they should've come away with a victory, but Sam Allardyce and Everton will be delighted with the result. Chelsea fail to capitalize on Arsenal and Liverpool dropping points, while Everton continue their resurgence under Allardyce and move up the league table as the festive period of football begins.