It looked like Jose Mourinho's side would move four points clear after defender Branislav Ivanovic, who got the winner in Chelsea's 1-0 triumph at Manchester City last week, smashed the ball home in first-half injury time to put the visitors ahead.
Anichebe, however, grabbed a precious point for the sturggling hosts when he headed in Saido Berahino's cross to move West Brom out of the relegation zone on goal difference.
Chelsea now have 57 points from 26 matches, two ahead of Arsenal, who host Manchester United on Wednesday, and three clear of City, who are at home to Sunderland.
At the other end of the table, West Ham United again showed they can win without suspended striker Andy Carroll as they beat Norwich City 2-0 to ease their relegation fears with a third successive victory and fourth straight clean sheet.
Second-bottom Cardiff City missed the chance to move out of the drop zone when they were held 0-0 at home to Aston Villa while Southampton jumped to eighth with a 1-0 win at Hull City.