The 26-year-old had been linked with some of Europe's top teams before completing a somewhat surprising switch to Bayern on a two-year loan with an option to buy.
The Colombia international reunites with former Madrid boss Carlo Ancelotti at the Allianz Arena, and he insists that he is "fired up" to get started.
"I'm very happy and looking forward to being at a huge club and in a great team. Everyone knows what FC Bayern stands for and what the club is about," he told the club's official website.
"It's a new culture, a new club, a new life. But I'm really fired up, I'm dreaming big, and I'm hoping for great success here."
Bayern are understood to have the option to sign Rodriguez in 2019 for a fee of around €35m (£31m).