Everton dug in to hold on for a nervy point after a tense and scrappy goalless draw against Crystal Palace at Selhurst Park - but it was not enough to prevent the Blues from dropping into the bottom three.
Leicester’s win over Wolves means Everton drop to 18th place on goal difference with six games remaining.
The Toffees had to play the final 10 minutes with 10 men after Mason Holgate was shown a second yellow for a late challenge on Jordan Ayew.
Holgate’s tackle was stupid - but in mitigation it was his first Premier League since August. Dyche arguably should have brought him off well before he risked making some tired challenges.
The headline news before kick-off was the welcome return of Dominic Calvert-Lewin to the starting line-up for the first time since early February.
It has been a nightmare few years for the England striker and Everton have suffered equally in his absence.
The Everton fans were also true to their word, arriving at Selhurst Park early to boisterously back the team, which they continued to do throughout.
It was a pretty cagey opening 45 minutes in the south London sunshine, with chances mainly coming from the distance.
Everton’s best opportunity was a fizzing first-time volley from Alex Iwobi that was pushed away by Sam Johnstone at full stretch. For Palace Jordan Ayew, who always seems to have a good game against Everton, header over from a dangerous Eberechi Eze free kick.
Eze had the ball in the back at the start of the second half when he ran onto a ball over the top, but his celebrations were cut short by the offside flag.
The Toffees were sinking back deep into their own half as the game wore on, but Calvert-Lewin had a wonderful chance to snatch the lead when he controlled Iwobi’s pass and fired just wide.
The lively Eze - who looks a brilliant player - arguably had the effort of the game with a curling shot from outside the area that was well tipped round by Pickford.
Holgate’s dismissal pretty much ended Everton’s attacking threat as they looked to preserve what they had.
It was tense, it was nervy, but the showed the kind of battling qualities needed if they are to fight their way out of trouble.
The issue, though, is whether there is simply enough goals in this team.