John Stones and Phil Jagielka, partners in central defence for the Toffees, both produced impressive displays as did Ross Barkley in midfield to help the Merseyside outfit claim the points in front of the watching Three Lions boss.
Steven Naismith, who came off the bench early on, scored a hat-trick as the champions’ struggles continued with their third loss in their opening five league games.
The super sub headed in England Under-19s international Brendan Galloway's cross before rifling in a shot to put the hosts 2-0 up at Goodison Park.
Jose Mourinho’s side responded with a thumping 30-yard strike Nemanja Matic to give them some hope, however, that disappeared late on as Naismith claimed the matchball and the victory with an angled drive.
Chelsea, who only lost three games in the whole of last season's successful league campaign, have now made their worst start to a top-flight season since 1986.
Stones, who handed in a transfer request earlier in the campaign, did not take long to show his mind was firmly on the task at hand. Having been nutmegged by Diego Costa, the 21-year-old quickly turned to get the ball before the Chelsea forward could get clear in the early exchanges.
His defensive colleague Jagielka then twice put in vital challenges to prevent the Spanish international attempts to attack.
"John has been heavily tested but it was never in doubt that he would perform well. To keep a clean sheet in the second half is down to the psychology of the group," Everton boss Roberto Martinez said afterwards.
Later on at the Emirates, England forward Theo Walcott was again in the goals as Arsenal beat Stoke City 2-0.
Walcott opened the scoring just after the half-hour mark when he chased a long ball over the defence from Mesut Ozil before firing low beyond international team-mate Jack Butland.
Theo Walcott notched his first of the season for Arsenal
The strike added to his brace for Roy Hodgson's men seven days earlier when they clinched their place at Euro 2016 with a 6-0 win in San Marino.
The man who Walcott came on for in Serravalle, Alex Oxlade-Chamberlain, was introduced with five minutes to go and it was a foul on him which resulted in the second moments later.
Fellow substitute Olivier Giroud sealed the victory with a header from Santi Cazorla's delivery.
Though he conceded two, Stoke stopper Butland was one of the game's stand-out performers and made no fewer than 10 saves which frustrated the home side for large spells.
After the game, Walcott said he was happy with his opener and believes he can be a regular for the Gunners in any of the front three positions.
"I was pleased with that goal," he told Arsenal.com. "I was disappointed because I should have got many, many more, but I’m in the chances so I’m happy with that.
"I’ve said to the manager that I’d like to play up front. I can play in any of those three positions up front, so it’s a good option for the manager to have."
Elsewhere, in the Championship’s lunchtime fixture Nottingham Forest came from a goal down to defeat QPR at Loftus Road.
R’s forward Charlie Austin, who received his first Three Lions call up in the summer, broke the deadlock to give the hosts the lead on 65 minutes.
Former England Under-21s midfielder Henri Lansbury levelled for Forest from the spot, sending past Alex Smithies after goalkeeper Robert Green, who has 12 England caps, was sent off for bringing down Chris O'Grady.
Substitute Nelson Oliveira then slotted in the winner for the visitors with eight minutes remaining.
Tyler Walker, who scored twice last Saturday on his England U20s debut in a 5-0 win over Czech Republic, started for Forest but was replaced at half time.
