Brazil's Richarlison headed Marco Silva's side in front from close range in the 49th minute after Kepa Arrizabalaga had kept out striker Dominic Calvert-Lewin's powerful header from Sigurdsson's corner.
Everton went 2-0 up in the 72nd minute when Sigurdsson tucked the ball in past Arrizabalaga after his penalty kick was saved by the Chelsea goalkeeper following a foul in the box on Richarlison by Marcos Alonso.
Chelsea had been unlucky not to score after a bright start. Eden Hazard's low drive was saved by Jordan Pickford in the sixth minute, before he cut in and hit the post a minute later and Argentina striker Gonzalo Higuain had a shot cleared off the line.
Defeat for sixth-placed Chelsea left them with 57 points from 30 matches, a point behind Manchester United above them and three points behind Arsenal in fourth.