Chelsea scored twice inside the opening three minutes to seal their third consecutive win but holders Manchester City were shocked at home by Stoke City and Manchester United were held in a goalless draw at Burnley
Diego Costa began the rout at Everton with a goal after 34 seconds and ended it in the final minute to take his tally to four goals in three games since joiningfrom Atletico Madrid over the summer.
Branislav Ivanovic added to Costa's early goal to leave Everton 2-0 down at Goodison Park but Kevin Mirallas reduced the deficit just before halftime. Seamus Coleman's own goal made it 3-1, with Nemanja Matic and Ramires also scoring for Jose Mourinho's side. Steven Naismith and Samuel Eto'o were Chelsea's other scorers.
Earlier, Mame Diouf scored a spectacular solo goal to give Stoke a surprising 1-0 win over Manchester City.
Diouf received the ball on the edge of his own area and ran the length of the pitch before slotting a low shot between the legs of Joe Hart, which the England goal-keeper should have kept out.
Manchester United's Premier League frustrations continued as they were held 0-0 at promoted Burnley with British-record signing Angel Di Maria unable to spark an improvement in their fortunes.
Di Maria, signed from Real Madrid for $98.77m on Tuesday, showed flashes of his talent, but United still looked short of attacking ideas, claiming only their second point from their opening three league games.
Their defence, that were ripped apart in a 4-0 League Cup defeat to third tier MK Dons on Tuesday, still looked uncomfortable with manager Louis van Gaal's three centre back system.
Newcastle 3 - 3 Crystal Palace
QPR 1 - 0 Sunderland
Swansea 3 - 0 West Brom
West Ham 1 - 3 Southampton