The league leaders looked set to extend their advantage at the summit to four points after Branislav Ivanovic – as he did eight days ago at Manchester City – gave them a first-half lead.
But the Serbian defender's contribution would not prove decisive on this occasion as Saido Berahino's 87th-minute cross was glanced home by substitute Anichebe, who also ensured a share of the spoils when Liverpool visited The Hawthorns last weekend.
Arsenal and Manchester City can both leapfrog Jose Mourinho's team with wins in their respective games against Manchester United and Sunderland on Wednesday night, while West Brom edge out of the relegation zone ahead of the latter club on goal difference.
Thievy Bifouma marked his West Brom debut at Crystal Palace on Saturday with a goal less than a minute after coming on.
The on-loan Espanyol forward was rewarded with a first start, one of four changes made by Pepe Mel, and faced up to a Chelsea defence still lacking John Terry due to a muscle strain.
There was little to unduly concern either backline in the opening 20 minutes, although Chelsea had established a controlled hold on possession.
In the 22nd minute, Willian found Samuel Eto'o inside the area and his cutback was scrambled away from the onrushing Ivanovic by a combination of Gareth McAuley and goalkeeper Ben Foster.
West Brom grew impressively into the game and Chris Brunt fizzed a low drive wide after Bifouma failed to control Berahino's knockdown.
A succession of Chelsea corners came to nothing until their final set-piece of the half – in the third minute of stoppage time – was touched on by David Luiz and his defensive colleague Ivanovic slammed home unmarked.
Foster made a sharp stop from Eto'o when the former Barcelona striker collected Willian's 50th-minute throughball.
The West Brom goalkeeper bettered that save as the hour approached, adjusting to keep out Willian's deflected shot after his defence backed away from the rampaging winger.
Eden Hazard beat the West Brom offside trap in the 68th minute and teed up Eto'o, but his final contribution before being replaced by Fernando Torres was to slash wide of the near post.
Emboldened by their visitors' profligacy, West Brom mounted considerable pressure in the final quarter of an hour – Bifouma and Brunt both pulling attempts off target from inside the area.
McAuley then rose unmarked to meet Brunt's 81st-minute corner and could not direct his header downwards.
But the pressure told as Anichebe found a way through and, when Torres left the field at full-time amid a heated confrontation with Foster, it was indicative of Chelsea's feelings towards an opportunity missed.
