The 29-year-old has signed a contract with the Premier League outfit after being let go from Bundesliga outfit Werder Bremen following the expiration of his contract.
Baggies boss Steve Clarke told the club's official website: "I'm delighted to welcome Markus to the club.
"He is an experienced striker who has scored goals in three of Europe's top leagues, as well as in the top European competitions, and is determined to prove himself in the Barclays Premier League.
"He had several other offers this summer but he made it clear that joining West Bromwich Albion was his first choice and we're really pleased we've been able to conclude a deal."
The club have confirmed that Rosenburg will wear the number eight shirt.