Anelka, 34, was available on a free transfer after a brief spell at Juventus, and he has penned a one-year deal after passing a medical at the Hawthorns.
“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 for everyone at the club,” first team coach Steve Clarke, who worked with Anelka at Chelsea, said.
"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 is a well-travelled striker with a reputation for hot-headedness that has cooled in recent years, although he was a key protagonist in France's infamous World Cup 'strike' that effectively ended his international career in 2010.
He started his career at Paris Saint-Germain before joining Arsenal for £500,000 as a teenager. A big-money but ill-fated move to Real Madrid followed, with the Frenchman embarking on a journeyman career that has also seen him play for Manchester City, Bolton Wanderers, Chelsea, Liverpool, Fenerbahce and China.