Everton climbed into the top four to stay on course for a European club competition berth next season.
West Brom's narrow defeat and their first loss in almost a month left them still deep in relegation trouble. They have 17 points from 27 games, nine points from safety, as they stayed second bottom of the table.
Brazilian Richarlison rose above the home defence at The Hawthorns to power home a 65th-minute header from a curling cross by Gylfi Sigurdsson, who had just come off the bench, for his sixth league goal of the season.
"We had an idea a set-piece might win it," Sigurdsson said. "We are confident after a couple of good results. But it is tight at the top. We have to keep going and work hard. If we get on a run we have a chance."
West Brom looked to have equalised for a much-needed point three minutes into stoppage time when Mbaye Diagne netted from close range, only for VAR to rule he was just offside.
It was a heartbreaking finish after they had started brightly with a good opportunity inside the opening two minutes for Diagne from Conor Gallagher's cross, only to be thwarted by a flying save from Jordan Pickford.
The Senegal international proved a real handful for the visiting defence and had several other chances.
Everton were largely on the back foot in the first half but despite limited opportunity were still able to conjure up chances of their own.
Everton looked more attacking in the second half, but still required strong defence to keep the home side at bay and take home all three points.