All the goals came in the first half of a clash between two clubs stuck at the wrong end of the table, with David Unsworth's hopes of getting the Everton job on a full-time basis boosted by a battling display on the road.
James McArthur popped up at the right time to score the first after a mere 51 seconds, only for Leighton Baines to convert a controversial penalty inside six minutes and make it 1-1.
After Wilfried Zaha tapped Palace back ahead, Roy Hodgson's side presented Everton with a second equaliser, Oumar Niasse scoring against the club he so nearly joined before the last transfer deadline.
The visitors improved in the second half but they were indebted to Palace substitute Christian Benteke for fluffing a glorious late chance to claim maximum points.
Ruben Loftus-Cheek was passed fit to start for the hosts and the midfielder – who came off in the first half of England's 0-0 midweek draw with Brazil with a back problem – was influential in Palace's opener.
The home fans weren't celebrating for long, however, as Niasse's theatrical fall when feeling contact from Scott Dann's outstretched arm convinced referee Anthony Taylor to point to the spot.
Leighton Baines slammed the ball home to draw Everton level with his 20th Premier League penalty, moving him just two behind manager Unsworth's record for a defender in the history of the competition.
Still, Palace responded well to the disappointment of conceding a dubious equaliser.
They produced the best move of the opening half to re-take the lead, Zaha on hand to provide the finish into an empty net after Joel Ward's low cross had zipped off the sodden surface to evade the diving Pickford.
Yet their hard work was undone by carelessness at the other end in first-half stoppage time, Speroni's unnecessary pass putting Dann in trouble. Idrissa Gueye robbed possession from the ponderous centre-back and teed up Niasse for his fifth of the campaign.
Speroni went some way to making amends for his part in the avoidable equaliser with a reactionary save after the break from Dominic Calvert-Lewin, one of two changes made by a clearly unimpressed Unsworth during the half-time interval.
The introduction of Calvert-Lewin and Tom Davies gave Everton some much-needed energy, albeit the latter was a little enthusiastic with a heavy tackle on Loftus-Cheek that resulted in a booking.
Hodgson utilised his bench to send on the fit-again Benteke for the final 15 minutes, with the striker given a brief cameo after missing 10 games through injury.
However, the Belgium international's lack of recent action came back to bite Palace when he was played clean through by Zaha, his scuffed shot failing to even trouble Pickford as it bobbled wide of the target.
Key Opta stats:
- Palace have collected five points from their opening 12 Premier League games – only one team has ever survived relegation having won five points or fewer at this stage (Everton, 4 pts in 1994/95).
- After failing to score in five of six home Premier League games between April and September, Palace have scored two or more goals in three consecutive top-flight home games for the first time since August 1991.
- James McArthur’s goal after 51 seconds was Crystal Palace’s second fastest goal in Premier League history – only Dwight Gayle against Newcastle (31 seconds) in August 2014 was scored quicker; it was also the fastest PL goal scored so far this season.
- Everton have conceded inside the opening minute of 10 different Premier League games – more than any other side in the competition’s history.
- Oumar Niasse has scored more Premier League goals this season (5 in 7 games) than he managed in his previous two seasons combined (4 in 22).