Anichebe struck with three minutes remaining after Albion had failed to clear a cross from Tony Hibbert.
It earned the Toffees their fourth away win of the campaign and allowed them to leapfrog Albion in the table.
The Baggies have won only two of their home league games this season and their tally of seven goals in front of their home fans remains the joint worst record in the top four divisions.
Both sides had struggled to carve out clear-cut openings before Anichebe struck, with Toffees keeper Tim Howard not having a shot to save in the entire 90 minutes.
Albion were the first to settle and winger Jerome Thomas got on the end of a low ball into the danger area from Billy Jones but lost his footing.
It was 18 minutes before the first reasonable opportunity when Shane Long headed just over after getting on the receiving end of a deep centre from West Brom skipper Chris Brunt.
Then John Heitinga produced a fine tackle inside the Everton area to halt the run of James Morrison.
West Brom keeper Ben Foster flapped at an overhead kick from Louis Saha following a Leon Osman corner.
But the former England keeper atoned for his error by rushing out to block the resulting shot from Jack Rodwell.
Everton looked the more likely to break the deadlock and Nicky Shorey blocked a fierce drive from Leighton Baines after his initial corner was half cleared.
Saha sent a half-volley just past the post after Albion failed to clear a Baines free-kick fully.
Everton defender Phil Jagielka screwed a shot just wide from another Baines centre.
Albion looked short of attacking ideas and Heitinga blocked a long-range attempt from Peter Odemwingie.
Rodwell should have done better with a free header from a Baines corner but he directed it wide.
Albion showed more purpose at the start of the second period and Odemwingie missed a clear chance after 47 minutes.
The Nigeria international stayed onside in running onto a looping header forward from Morrison and got clear of the Everton rearguard.
But Odemwingie lacked composure and lifted his shot over the bar.
Jagielka did well to cut out a low Jones cross with Long waiting in space behind him to take advantage.
Heitinga managed to nick the ball away from Odemwingie as he prepared to shoot inside the Everton area.
Morrison floated a pass over the Everton defence to Brunt but Sylvain Distin had read the move and was able to get back and block his shot.
Foster saved a volley from Everton substitute Magaye Gueye at the foot of a post.
Then, with three minutes left, Anichebe netted the winner.
Osman's neat flick found Hibbert in space and Scharner failed to deal with his cross. The ball broke to Anichebe, who slotted home his first league goal of the season.