This was not supposed to happen. While the away side are in decent form, they remain inconsistent in their travels, not scoring in five of their past seven away trips.
Indeed, this triumph secured their first back-to-back away wins in the Premier League since 1997.
In contrast, Everton were on a run of seven straight victories, their most consistent spell of form since 1987.
So, how did this happen? With the aid of some film, stats and graphics, here's a breakdown of the Toffees' failings.
Too Much of the Same in Attack
Roberto Martinez knew the pattern of this game. Everton, desperate for a win, would attack a densely populated final third, while Palace looked to condense the space.
To create as much room as possible, he selected a line-up of explosive talents capable of stretching the field, beating a man and creating an opening.
Kevin Mirallas, Gerard Deulofeu, Aiden McGeady and Ross Barkley can all be devastating on their day; starting them all together, however, is far too one-dimensional.
As this clip shows, too many players were looking to do the same thing.
This meant Everton rarely ventured between Palace's lines to move defenders around or disrupt their opponent's rigidity.
Instead, the Toffees hugged the touchlines and looked to make an impact from out wide, which played into Palace's hands.
Teams need one or two such explosive threats—not an entire attacking quartet.
Despite 65 percent possession and numerous take-ons in dangerous areas, the Toffees managed just one shot on target by half-time.
Individual Flair over Team Cohesion
Everton average just under 22 dribble attempts per game under Martinez.
As a consequence of this line-up, that number reached 36 in this match—the Toffees' second most of the season.
Everton made a staggering 36 take ons vs Crystal Palace, their second highest EPL tally of the season (41 vs Fulham). pic.twitter.com/VKYncYzoNd— Squawka Football (@Squawka) April 16, 2014
While 18 were successful, this is an indication of Everton's limitations in attack and further emphasises the previous point.
Martinez's side also sent in 46 crosses, their most of the season and double their average return of 23.
The fact that over half of these (25) went in during the first period suggests an element of recklessness about the Toffees' early play.
This is consistent with Martinez's post-match assessments, via the team's official website:
The result is a hurtful one because of the expectation that we had and the desire that we brought into the game.
In the first half we played with a bit of wanting to win so much that we forgot about simple basics.
Going behind clearly panicked the Toffees, who lost all sense of unity for most of the first half.
Everton's attackers were stuck with their heads down, looking to beat a man, cross or shoot instead of looking up and finding a way through as a team.
There were few combinations, and this quickly became far too individualistic. More patience, guile and team interplay was required to play through what was a well-drilled defence.
Attackers Slow to Help Out in Defence
With Everton's attack reduced to flashes and spurts instead of sustained pressure, Palace found several opportunities to counter-attack.
Again, this was helped by Palace's line-up. As this graphic shows, Everton's average positions reveal an almost 5-0-5 formation, with little in the shape of a midfield.
Deulofeu, McGeady, Mirallas and Barkley are hardly famed for their defensive contribution, and this left Gareth Barry overrun in midfield.
As this next clip shows, Palace were able to commit bodies forward and overload the Toffees' back four—something Martinez perhaps didn't predict them doing so aggressively.
Without James McCarthy to press alongside him, Barry was often left chasing the ball instead of holding his position—as was the case for Palace's opening goal.
Another angle shows just how many bodies were stuck upfield as Tony Pulis' side grabbed a crucial lead.
This become the tone for the first half, and the away side were unfortunate not to grab a second goal in a similar manner, as Cameron Jerome's shot came back off the post.
What was most to blame for this defeat?
Right Changes but All Too Late
Martinez has made very few errors during his time at Everton, and, to be fair to the Catalan, he changed things for the better after half-time.
Steven Naismith came on to provide another presence between the lines as well as in the box.
Leon Osman and McCarthy also arrived shortly before the hour mark, putting better pressure on the ball-carrier and providing more craft and cohesion in attack.
From the second-half showing, Everton were unfortunate not to at least claim a point.
They fired in 12 shots to Palace's three but drew the second period 2-2, let down by some more sloppy defending.
This will come as a deflating loss to the Toffees, who must quickly rouse themselves for David Moyes' visit with Manchester United on Sunday.