Rovers announced Santa Cruz has joined for "an undisclosed fee" but reports in Germany suggest the Premier League club have paid £3.5million to end his eight-year stay with the Bundesliga side.
Bayern have allowed Santa Cruz to leave following the summer arrivals of Luca Toni and Miroslav Klose, with coach Ottmar Hitzfeld recently confirming he was looking to sell the South American.
Blackburn boss Mark Hughes, himself a former Bayern Munich striker, has won the race.
Hughes said on Friday: "Santa Cruz is a young player, an international with a good reputation who is playing at a top European club.
"The good thing from our point of view is that his fitness levels are fine as he has been playing in the Copa America."
Hughes is already well-off for strikers, with Benni McCarthy, Jason Roberts, Matt Derbyshire and new recruit Maceo Rigters all vying for a first-team place.
In his eight years with Bayern, Santa Cruz has won the Champions League and five Bundesliga titles, as well as four German Cup triumphs and a World Club Cup winner's medal.
In 155 games, he scored 31 times and will bring a wealth of European experience to Rovers too, having played 51 games in the Champions League and scoring seven goals.
Vi bruger cookies for at kunne give dig den bedste oplevelse. Ved at bruge vores side accepterer du brugen af cookies. Du kan læse mere i vores persondatapolitik.