Chelsea passed up a number of chances to register in the first period and it proved to be Everton's day as the Toffees scored twice without reply in the second 45 minutes of action.
Chelsea had three huge opportunities to take the lead in the first 10 minutes; first Everton stopper Jordan Pickford did well to keep out a low strike from Eden Hazard, before the Belgium international hit the post with a curling effort moments later.
Gonzalo Higuain was next to come close for the visitors, but a combination of Pickford and Yerry Mina - starting a league game for the first time in 2019 - managed to keep the ball out of the back of the net.
Chelsea were the team having all of the opportunities and Jorginho was next to try his luck in the 20th minute, but Pickford was once again on hand to make a decent stop, before Ross Barkley - booed throughout the afternoon on his return to Goodison - was denied by the Toffees goalkeeper.
Everton were finding it difficult to create openings, but Chelsea stopper Kepa Arrizabalaga did have to keep out a strike from Andre Gomes in the 26th minute following a quick break from the home side.
Chelsea had another super chance in the 30th minute when Pedro found space inside the Everton box, but the Spaniard fired his effort wide of the post as the hosts survived another dangerous moment.
Higuain then clipped over the Everton goal from close range, before Dominic Calvert-Lewin headed a deep Sigurdsson free kick over the crossbar down the other end of the field.
The opportunities continued to arrive late in the first half and Pedro actually had the ball in the back of the Everton net in the 44th minute, but the linesman's flag came to the home side's rescue.
Everton started the second period on the front foot, though, and Bernard came close to meeting a whipped cross into the Chelsea box before Kepa had to keep out a powerful strike from Gomes.
The Toffees did take the lead in the 49th minute, however, when Richarlison headed home from close range. Kepa had initially kept out Calvert-Lewin's effort, but Richarlison was on hand to nod into the back of the net.
Marcos Alonso smashed one into the side-netting in the 56th minute as Chelsea looked for a response, but Everton so nearly doubled their lead just before the hour; a wonderful cross from Lucas Digne found Calvert-Lewin unmarked, but the Englishman could not make proper contact.
Pickford then had to keep out a powerful Higuain drive before Michael Keane blocked a goal-bound attempt from Barkley as the pressure from the capital side continued.
Everton were gifted a penalty in the 71st minute when Alonso brought Richarlison to the deck with a lazy tackle.
Sigurdsson stepped forward and saw his spot kick saved by Kepa, but the ball broke kindly for the Iceland international, who slotted the rebound into the bottom corner to double Everton's advantage.
Theo Walcott had a fine chance to make it 3-0 10 minutes from time, but the substitute could not find a route past Kepa as Chelsea continued to come under pressure.
Callum Hudson-Odoi saw his powerful strike saved by Pickford in the 82nd minute as the visitors threatened a late response, but Everton closed out the final stages to collect all three points courtesy of a two-goal success.
EVERTON (4-2-3-1): Pickford; Coleman, Keane, Mina, Digne; Gueye, Gomes (Schneiderlin 68'); Bernard (Walcott 78'), Sigurdsson (Davies 86'), Richarlison; Calvert-Lewin
CHELSEA (4-3-3): Kepa; Azpilicueta, Rudiger, Luiz, Alonso; Kante, Jorginho (Hudson-Odoi 74'), Barkley (Loftus-Cheek 65'); Pedro, Higuain (Giroud 65'), Hazard