There were only three minutes of normal time left when substitute Anichebe scored the winner in only his second league appearance this season.
Leon Osman's flick found Tony Hibbert in space and West Brom's Paul Scharner failed to deal with the ensuing cross.
Nigeria striker Anichebe, whose career has been plagued by injuries, then showed good strength to hold off several defenders and no little skill in then lofting the ball over a crowd of legs.
West Brom were desperately disappointing in front of their own fans, failing to register a single shot on target although Peter Odemwingie struck a dipping volley narrowly over the crossbar.
Victory for visitors Everton saw David Moyes's men climb to ninth place in the table and dropped West Brom down into 12th spot.
Source: AFP
Source: AFP