Chelsea were poised to move four points clear at the top of the table on the back of defender Branislav Ivanovic's injury-time strike in the first half at The Hawthorns.
But substitute Anichebe flicked home his third goal of the season in the 87th minute to make it 1-1 and thwart the Blues, who had lost on their last two visits to this part of the West Midlands.
After all the hype in the wake of their stunning win at Manchester City and Eden Hazard-inspired thumping of Newcastle on Saturday, Mourinho's side were undoubtedly off the boil against the Baggies.
Perhaps Mourinho does have a point after all, that his side are the "little horses" in this year's championship race, as Arsenal and City will have the chance to take over at the summit on Wednesday.
Only Willian can claim to have had an impressive game as it was he who primarily pulled the strings, and who crucially set up the only goal of the first half just as Albion were on the brink of heading into the interval on the back of a creditable 0-0 scoreline.
From Willian's corner into the near post, centre-back David Luiz was left unmarked to flick the ball on with his right foot to the far post where Ivanovic slotted home from three yards.
After the break Willian enjoyed a further three driven efforts, all of which were superbly saved by home goalkeeper Ben Foster.
At least the home side showed greater attacking intent in the second period as James Morrison fired wide, Thievy Bifouma flashed a shot through the six-yard box, while skipper Chris Brunt hammered a half -volley beyond the left-hand post.
With Anichebe on for Bifouma in the 73rd minute, the Nigerian glanced home Saido Berahino's cross three minutes from time to lift Albion back out of the relegation zone.