Everton given helping hand as Richarlison penalty extends perfect start
Carlo Ancelotti's side took the lead through Dominic Calvert-Lewin's fifth goal of the campaign in the first half at Selhurst Park.
Cheikhou Kouyate equalised for Palace before Eagles defender Joel Ward was penalised for handball after the video assistant referee advised Kevin Friend to consult the pitchside monitor.
Brazil forward Richarlison converted the spot-kick and Everton held on in the second half to make it three successive league victories and five wins in a row in all competitions.
Everton sit top of the table at this early stage and while a title challenge is unlikely, they could be contenders for a top-four finish.
The visitors went ahead with their first serious attack in the 10th minute.
Colombian playmaker James Rodriguez unfurled a sublime pass into Seamus Coleman and the captain cut back for Calvert-Lewin to net with conviction.
Palace hit back in the 26th minute when Kouyate got between Michael Keane and Richarlison to head his second goal for the club.
The Eagles richly deserved it and yet they were almost immediately pegged back when VAR had to look at a handball incident involving Ward on the half-hour mark.
Richarlison's cross had hit the Palace full-back on the shirt sleeve and it was not deemed a spot-kick, but it was only a brief let-off for Roy Hodgson's men.
In the 40th minute, James's superb pass was headed by Lucas Digne onto Ward's hand in the penalty area.
Friend pointed to the spot after using the pitchside monitor and Richarlison fired into the corner to put Everton back in front.
Calvert-Lewin should have grabbed his sixth goal of the season and put the game to bed in the 51st minute, but his effort from James's corner sailed wide.
As rain poured down in south London, Palace pressed for an equaliser, with Wilfried Zaha and Ebere Eze having shots blocked.
Eze was replaced by Michy Batshuayi with 15 minutes left and Hodgson also introduced Christian Benteke, but it was to no avail.