Ten-man Everton grimly hung on to secure a vital point, holding off a late charge from their hosts after Mason Holgate was sent off after 80 minutes.
A largely dour encounter was livened up at the death when the full-back was dismissed for a second bookable offence which led to a late flurry from the home side that could not yield a winner.
Everton welcomed back Dominic Calvert-Lewin, who lasted for 90 minutes and embodied a determined effort which rarely quickened the pulse and does not massively improve the situation they find themselves in.
For Palace it pushed them to 37 points and it looks all but mission accomplished for Roy Hodgson in keeping them up.
Calvert-Lewin was handed a first start since facing Arsenal in early February, Dyche’s first game in charge. His availability coming as a pre-match morale booster for the visitors, who had put the striker through his paces midweek by giving him an hour in a friendly against Chester.
Everton also brought in Holgate for Ben Godfrey as they tried to break the sequence of going thirteen games without a win in all competitions.
Palace, hoping to notch a fourth straight win in a row for the first time since June 2020, were happy to stay with an unchanged eleven, charged with continuing the surge that had been enjoyed under Roy Hodgson.
‘Fight like your fans,’ extolled a banner in the away section, as Calvert-Lewin – watched by Gareth Southgate from the stands - found himself as a lone front man with Alex Iwobi in a free-roaming role behind him.
The restored striker’s first significant touch came in the eighth minute when Iwobi found him in the box, but the header was not enough to trouble Sam Johnstone.
Most of the action was coming at the other end with the home side’s crosses from the left flank causing problems and Ayew heading over as the Everton defence began to creak.
On the other side Vitali Mykolenko was also having problems against Michael Olise, who he needed to manhandle off the ball before a cry for a penalty was turned down.
While Everton scratched around Palace were playing with a free-flowing confidence of a team hauling themselves to safety, although there was little in the way of clear cut chances to show for it.
Everton’s first opening came on the half hour with Iwobi’s volley from the edge of the box flying through an array of bodies and producing a fine diving save from Johnstone. The midfielder was normally involved in the best stuff that came from the away team, and that heralded an improved spell for the visitors, who will have entered the break feeling they had weathered the initial storm.
Calvert-Lewin offered a physical presence – clattering into Joachim Andersen several times – and a focal point for balls into the box but not much specific threat.
On 55 minutes the recently prolific Eberichi Eze chipped Pickford and rolled the ball into the net after being put through by Marc Guehi, but it was clearly offside.
Five minutes later the Everton centre forward was back in it when he turned beautifully onto Iwobi’s through ball onto his left foot, only to slide it past the post from twelve yards out.
With fifteen minutes left Pickford needed to parry away the side-footed attempt of Eze, but the home side were struggling to recreate the buzz of their opening quarter.
Their hopes of seizing victory would be aided when, on 80 minutes, the struggling Holgate was dismissed for his second yellow card offence when he upended Jordan Ayew after the striker had zipped past him down the left flank.
That swung all the momentum in Palace’s favour and soon Dyche’s men were having to fling bodies around in the box to save themselves as Palace forced a string of corners.
UItimately they were able to come through the storm of the last few minutes to go away with a point from a largely forgettable encounter.