The capital club made the better start of the two teams as they dominated possession but a different Everton side came out after the break and it was not long before the hosts were ahead at Goodison Park.
The first goal came in the 49th minute from an Everton corner which was headed at goal by Dominic Calvert-Lewin before being saved by Kepa Arrizabalaga, only for Richarlison to latch onto the loose ball and open the scoring with a close-range header.
The Brazilian had failed to find the net against any of the established top six before that goal against the west Londoners.
Richarlison had a hand in the second as he was brought down by Marcos Alonso in the box with referee Anthony Taylor awarding the penalty.
Sigurdsson saw his initial shot from the spot well saved by Kepa but the rebound fell straight back in the path of the Icelander who tapped home an easy finish.
Chelsea did have chances to get back into the game, with Callum Hudson-Odoi forcing a good save from England keeper Jordan Pickford.
However, the visitors could not break the Everton defence down and it was Marco Silva’s side who claimed the three points to move up to 11th in the Premier League table.