Rovers announced Santa Cruz has joined for "an undisclosed fee" but reports in Germany suggest the Premier League club have paid £3.5million for his services.
Rovers chairman John Williams believes the club have shown shrewd business sense by signing Santa Cruz.
Williams told rovers.co. uk: "We set off with the intention of securing quality at the right price and I think we have achieved that."
Blackburn boss Mark Hughes said: "We should not underestimate the football grapevine.
"People are beginning to understand what we are trying to do here. Word gets around and that gives you a foot in the door.
"Obviously the deal has to be right for the individual and the club. I am sure Roque will enjoy the day-to-day working here.
"I am delighted to get him. He brings quality and great technical ability. He is going to complement the very good strikers I have here.
"Competition is always good. I am very excited at what is in store."