The French striker returns to the Premier League on a 12-month contract following an unhappy year in Chinese football and a short spell in Italy, and Albion have the option of awarding him a further year's deal next summer.
Anelka, 34, left Chelsea for Shanghai Shenhua in January 2012 but joined Juventus during this year's transfer window.
A move to The Hawthorns reunites him with Steve Clarke, the Albion boss who coached him briefly at Chelsea in 2008.
Clarke, who has seen last season's top scorer Romelu Lukaku return to parent club Chelsea, said: "We are missing a bit of firepower from last year - we're trying to put that right - and to kick off with the signing of Nicolas at the start of the summer is a great boost.
"The more quality players you can bring to the club the better - and no-one can question Nicolas' quality.
"He's a player I have worked with before as I was at Chelsea when they signed him. He's got a great work ethic, looks after himself and is a consummate professional.
"If you add that to the obvious talent he's got I think he can prove to be a really top signing for the club.
"His experience with some of the top clubs in England and Europe will stand the group in good stead.
"Hopefully he will be able to inspire the other players to even greater heights than we managed to achieve last year."
Anelka, 34, has played for 10 clubs in his career and Albion are his sixth in England.