The Baggies forward headed smartly home in the 87th minute to cancel out Branislav Ivanovic's first-half opener and leave the door open for Arsenal to retake their lead
The Blues looked set to extend their advantage at the top of the table to four points after Branislav Ivanovic - as he did eight days previously at Manchester City - gave them a first-half lead.
But the Serbian defender's contribution did not prove decisive 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 earlier in February.
Arsenal and Manchester City could both leapfrog Jose Mourinho's team with wins in their respective games against Manchester United and Sunderland on Wednesday night, while West Brom edged out of the relegation zone ahead of the Black Cats on goal difference.
Thievy Bifouma, who marked his Baggies debut at Crystal Palace on Saturday with a goal less than a minute after coming on, was 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 back line in the opening 20 minutes, although the Blues 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 Thievy 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 through-ball and 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 hosts' 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 - Thievy and Brunt both pulling attempts off target from inside the area. McAuley then rose unmarked to meet Brunt's 81st-minute corner but 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.