Frank Lampard's Chelsea -- who were unbeaten in their nine previous games -- had the chance to reach the summit of the Premier League table but a resolute Everton defence and a Gylfi Sigurdsson penalty saw the visitors miss their chance.
A clumsy penalty against Chelsea keeper Edouard Mendy on Dominic Calvert-Lewin gifted Everton their opener as Sigurdsson stepped up and dispatched the spot kick to make it 1-0 in the 22nd minute.
The hosts had Jordan Pickford to thank for keeping Chelsea scoreless soon after their opener, the England international making a nice diving save from a Reece James free kick and gobbling up Kurt Zouma's effort from close range following a set piece.
Trailing by a goal after the break, Chelsea's Mason Mount had a golden chance early in the second half go begging when he was unable to put the ball into the net after Pickford had been caught off his line.
Calvert-Lewin looked to have drawn a second penalty for Everton when he was pulled down by Ben Chilwell in the area, but the flag had gone up for offside on the ball that played him through and Chelsea were awarded a free kick.
Mount was unlucky not to score in the 80th minute from a free kick, as his shot had beaten Pickford but came back off the far post.
"It was a tough one, not going to lie but we had to defend until the last second. A very important three points for us," goal-scorer Sigurdsson told BT Sport.
"Against a team like Chelsea, they are good going forward and move the ball really well. We have done some work on our defensive shape during the week and it worked."
Everton, who have been a bogey team for Chelsea at Goodison Park, clung to their lead to pick up all three points to move to 20 from 12 games on the season. Lampard's Chelsea are still in third on 22 pending the outcome of Sunday's matches.