Wasteful West Brom extended their winless run to 18 games after failing to find a way through against Everton
Salomon Rondon and Craig Dawson missed golden chances as the Baggies were held to a goalless draw at The Hawthorns, with boss Alan Pardew still looking for his first win after six attempts.
They will go half a season without a Premier League victory if they fail to beat Arsenal at home on Sunday and Pardew's side are three points from the safety line.
That Albion are not cut adrift will be a crumb of comfort but they needed Ben Foster's late save from Oumar Niasse to clinch a point.
Wayne Rooney was forced to return to Merseyside after failing to recover from a virus having travelled to the midlands on Christmas Day but Yannick Bolasie did make his first Everton appearance for a year following a serious knee injury.
Without 11-goal Rooney though the visitors were toothless and relied on Albion's poor finishing to extend their unbeaten run to eight games.
The hosts had scored 14 goals in 20 games ahead of the Boxing Day fixture and should have buried the Toffees in the first half, only to continually let them off the hook.
First, after five minutes, Rondon wasted a glorious opportunity when he failed to connect with Jay Rodriguez's cross just five yards out.
The opening at least gave Albion belief and the busy Chris Brunt and Rodriguez also went close before Dawson should have given the hosts the lead after 19 minutes, only to power a header over from six yards.
The Baggies were more expansive and in control but lacked the security of a goal, even if Everton offered little threat to Foster.
Brunt's miscued effort was gathered by Jordan Pickford three minutes before the break and Gareth Barry's header landed on the roof of the net.
Albion's profligacy continued four minutes into the second half when Rondon sliced wildly wide and the striker's afternoon ended shortly after when he limped off with a hamstring problem.
It added to the Baggies' issues up front and, while they continued to force the issue in the second half, Pickford was never forced into a serious save.
Matt Phillips fired a free-kick over and Kieran Gibbs miscued when well placed before Brunt fired over.
But Everton almost completed a smash-and-grab win with two minutes left but Foster saved Niasse's low effort.