Ronald Koeman faces more difficult questions after Jeff Hendrick bolstered Burnley's impressive away record by sealing a 1-0 victory at Goodison Park.
Republic of Ireland international Hendrick was on the end of a sweeping move as the Clarets added to a collection of positive road results featuring a victory at Chelsea and draws at Tottenham and Liverpool.
The goal plunged Everton straight back into turmoil, with last weekend's win over Bournemouth -- their only league victory since the opening day of the season -- a distant memory as thy sank to a fourth league defeat in five matches to leave them just two points above the bottom three.
Burnley's 12-point haul represents their best start to a Premier League campaign and was sealed when Hendrick easily side-stepped Morgan Schneiderlin in the penalty area to slide home a low shot after a 24-pass move bettered only in numbers by Pep Guardiola's Manchester City this season.
Everton crumbled, with Ashley Williams lucky to escape when his clearance rebounded off Hendrick to Arfield, who was booked for diving in the penalty area after the pedestrian centre-back attempted to correct his error.
Koeman had criticised his players for being "afraid" after the midweek Europa League draw at home to Apollon Limassol and injected some youthful exuberance with Dominic Calvert-Lewin and Nikola Vlasic given starts.
Oumar Niasse, ineligible in Europe after being excluded from the squad, made his first league start for Everton since May 2016 and his first under Koeman.
He partnered Calvert-Lewin up front, with Wayne Rooney dropped to the bench for the first time since his return from Manchester United.
For a while it appeared to be working, with Niasse's energy supplemented by some purposeful running from the two youngsters, creating two chances for Gylfi Sigurdsson. But the Iceland international -- so clinical for Swansea but yet to open his Everton account in the league -- wasted them both.
However, Sean Dyche's Burnley were able to cut through them with ease midway through the first half.
From that point, the Toffees laboured with little reward, with Rooney and academy graduate Tom Davies sent on for the final quarter to try and break a well-drilled Burnley side's grip on the game.
That they failed was as much down to the Clarets' resilience as it was to Everton's failings.
Rooney thought he should have had a penalty when his flick-on from Sigurdsson's cross bounced onto Matthew Lowton's arm but, unlike last week, there would be no late reprieve.
The boos ringing out at the end left Koeman in no doubt about how bad a start this has been.