Chelsea was poised to move four points clear at the top of the standings after defender Branislav Ivanovic's stoppage-time strike in the first half at The Hawthorns.
But Anichebe headed home his third goal of the season in the 87th minute to thwart Chelsea and lift West Brom out of the drop zone on goal difference.
For 60 minutes there was only one team, and that team couldn't kill the game, especially in the first 15 minutes of the second half," Chelsea manager Jose Mourinho said. "For (the next) 10 minutes the game was poor, and then for the last 20 minutes only one team and that team fought a lot to get the goal and the point, and in the end they deserve it."
Manchester City also has a chance to move to the top by beating Sunderland if Arsenal slips up.
This was a night when the hype following Chelsea's wins over Manchester City and Newcastle was punctured. Only Willian can claim to have had an impressive game, primarily pulling the strings and setting up Chelsea's goal.
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