The 34-year-old joined the Baggies on a one-year deal after completing a medical at the club this afternoon.
Clarke hopes that he will able to replace some of the goals lost from his side last season when Romelu Lukaku returned to the Blues, following his successful loan spell at The Hawthorns last season.
He told the club's official website: "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.
"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's quality."
Anelka spent part of last season on loan at Juventus after leaving Chinese club Shanghai Shenhua.