Second-half substitute Rafael Garcia stung the palms of Whitworth after spinning on the ball inside the box, mustering what proved to be Everton’s only chance after pulling a goal back.
After seven minutes of stoppage time, the Eagles saw off the toothless Everton efforts and secured their first victory of the season.
Palace: Whitworth (GK), Robertson (Taylor 78), O’Brien, Jach, Hannam, M.Boateng, Roles, Omilabu, Wells-Morrison, Rak-Sakyi (Gordon 66), Street (Quick 66).
Subs not used: Webber (GK), Rich-Baghuelou, Taylor.
Everton: Leban (GK), John, Campbell, Astley, Welch, Butterfield, Mills, Whitaker, Dobbin, Price, Hughes (Garcia 60).
Subs not used: Barrett (GK), Cannon, Anderson, Higgins.