The 34-year-old, who was a free agent, signed an initial one-year contract at The Hawthorns on Thursday, with the club confirming on Twitter: "Albion have completed the signing of @anelkaofficiel on a one-year deal, with a further year's option in the club's favour."
Baggies head coach Clarke, who worked with Anelka for a short time at Chelsea, is delighted with the signing.
"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," he told the club's website.
"The sole focus at the club is trying to build on what we did last season.
"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 will wear his favoured number 39 for the Baggies - his sixth Premier League club after spells with Arsenal, Manchester City, Bolton, Chelsea and on loan at Liverpool.
He signed for Shanghai Shenhua at the start of last season but finished the campaign back in European football with Juventus.
He has also played for Paris Saint-Germain, Real Madrid and Fenerbahce and won 69 caps for his country, scoring 14 goals. He has not played for Les Bleus since the 2010 World Cup.