The Blues looked set to open up a four-point lead at the top of the Premier League table when Branislav Ivanovic swept home following a David Luiz flick from Willian's corner.
West Brom keeper Ben Foster saved well from Samuel Eto'o and Willian.
But the Baggies moved out of the bottom three after Anichebe got ahead of Luiz to head home Saido Berahino's cross.
Chelsea created enough chances to have won comfortably but will be replaced at the top of the table if either Arsenal or Manchester City win on Wednesday. The Gunners are at home to Manchester United, while City host Sunderland.
Anichebe's 87th-minute equaliser will raise questions as to whether Jose Mourinho's side, who were held to a goalless draw by West Ham two weeks ago, have the quality to win the title.
In a game that will not live long in the memory, Ivanovic's goal in the third minute of stoppage time was reward for a dominant first-half display.
Chelsea smothered West Brom with possession yet Willian's long-range effort was the closest they came before their Serbia defender pounced from inside the six-yard area.
Eto'o had already forced a fine save from Foster, while the lively Willian was denied a chance to open a two-goal cushion by the Baggies keeper.
West Brom, who came from behind to force a 1-1 draw against Liverpool on 2 February, rallied and again displayed their battling qualities.
But skipper Chris Brunt and James Morrison lost their composure when presented with chances to equalise.
And 21-year-old striker Thievy Bifouma, making his first start since joining on loan from Espanyol, fired across the face of goal.
That looked it for the hosts but, moments after Blues defender Gary Cahill had been treated for an injury, Anichebe sent Baggies fans into raptures with an excellent header that lifted his side out of the relegation zone into 17th spot.
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