The 25-year-old international signed a four-year contract after passing a medical earlier in the day, the Bundesliga club said on their website.
"It's done," said the club. "Arjen Robben will strengthen the Bayern squad with immediate effect."
They said Robben would train later on Friday and would be available for Saturday's match at home to champions VfL Wolfsburg, when Bayern will attempt to chalk up their first win of the season.
Under new coach Louis van Gaal, Bayern have taken only two points from their first three games in their worst start to a season for 43 years.
Their form has already brought stinging criticism from former players Oliver Kahn and Stefan Effenberg.
GOOD FEELING
"It all happened very fast. There were a couple of things I wanted to think over but in the end I decided I really wanted to move to a huge club like Bayern," Robben told Bayern's television channel FCB.tv.
"Right from the start, Bayern gave me a very good feeling, and I feel very welcome here. This is a massive club with a glorious history and I'm delighted to be here now.
"I'm totally fit, I had a good pre-season with Real Madrid and I'm ready to play," he added.
Van Gaal said: "I'm delighted, because now I have another creative player. He's an explosive player.
"I don't think we'll only be using a 4-3-3 formation from now on. We could use a number of systems with Robben and (Franck) Ribery."
Bayern did not disclose the transfer fee but Spanish media estimated they had agreed to pay 25 million euros.
Robben made his debut for Real in September 2007 after moving from Chelsea but despite a number of scintillating performances last season his career has been blighted by injury.
The arrival of Cristiano Ronaldo and Kaka at the Bernabeu had made it less likely that he could command a regular place in Manuel Pellegrini's team this season.
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