Alan Pardew recokns Albion are right in the relegation mix after the Baggies were unlucky to leave Goodison Park with only a point.
The Baggies impressed throughout and the hosts had Jordan Pickford and the woodwork to thank for restricting the visitors.
Jay Rodriguez gave Albion the lead inside seven minutes when he ran on to Grzegorz Krychowiak excellent through ball.
Everton equalised in the 70th minute when Oumar Niasse scored with his first touch.
The draw leaves Albion only three points from safety - and sixth from tenth.
"The last three games have got us points but the momentum started before that," Pardew said after the game.
"We're there. We're right in the mix. This point will hopefully come in crucial for us going forward.
"I have to say it's a great point for us because coming here is difficult. I don't know what the figure was on how much they've spent this week, £40m or something, but we subdued all their offensive players. They were all fit and available today.
"There's a little feeling of disappointment in the dressing room. That's a good feeling to have and we'll take that to the next game. You only have that feeling when you're playing okay."