James McCarthy's dismissal early in the second half presented a fine opportunity to secure the three points that would have taken their season's total to 40 and almost certainly have secured survival, but instead they remain on 38, and in 16th place, ahead of difficult fixtures with Arsenal and Manchester United.E
verton have so often been accused of disappointing and being poor in defence, but on this occasion a 0-0 draw represented success, even if it owed much to an inspired performance from Joel Robles and left them in 12th with 40 points.
After Saturday's 1-0 defeat of Norwich had given them their first league victory of 2016, Palace manager Alan Pardew had spoken of his hope that his players would shed the nerves he believes have recently undermined them, and so early on it looked likely to prove that way.
Jason Puncheon, who scored Saturday's winning goal, demonstrated improved confidence in only the third minute with a fine through-ball to Yannick Bolasie before the forward, from close range, tested Robles with a near-post shot.
Dwight Gayle was unexpectedly retained over Connor Wickham and Emmanuel Adebayor and he too threatened two minutes later when stretching to stab a first-time shot from Wilfried Zaha's masterful pass just wide of the right post.
Everton, arguably the season's biggest underachievers, then began to respond by creating convincing chances of their own
First Seamus Coleman produced a reaction save from Palace goalkeeper Wayne Hennessey with a powerful near-post shot, then Arouna Kone pounced on a loose ball to strike just over the crossbar.
In the 29th minute, Romelu Lukaku was played in by Ross Barkley, but watched his hooked, left-footed shot force Hennessey into another save.It all said so much about these two teams
There is no shortage of talent or creativity, but that lack of conviction meant that at half-time they each remained goalless when they both should have scored at least once.
Sensing his team required change, Pardew replaced Gayle with Wickham at half-time, but it was Everton who first threatened, when from the edge of the area Barkley evaded Yohan Cabaye's challenge to create the space from where he classily curled against the crossbar.
If such ability made Everton the favourites, however, the dismissal of McCarthy in the 52nd minute gave Palace a significant advantage
The midfielder had already been booked for a first-half foul on Cabaye when he needlessly blocked Bolasie to earn a second yellow card.Given Everton have consistently proved vulnerable in defence, the loss of one of their more balanced players proved a concern
This was highlighted when Mile Jedinak ran through the back four, found Puncheon, and watched as the forward wasted an outstanding chance when Robles saved.
Puncheon again threatened in the 67th minute, though while he this time created his own space and shot from a more difficult position, Robles produced another athletic dive to again deny him.
Adebayor came on for Bolasie, and soon sent a close-range header wide
He also created a promising chance Wickham could not connect properly with, and headed over late on, but the goal Palace needed ultimately eluded them.