"I am here to win titles. That is how it has always been in my career," said the 24-year-old playmaker as he was presented to thousands of fans and media at the Santiago Bernabeu stadium.
"When I started at Bayern Munich the only option was to win, and now it is the same," he added.
"I know that I am at a very successful club and I am going to do my bit to make sure that continues to be the case."
Spanish media said Real and Bayern agreed a deal worth between 25 million and 30 million euros ($33-$40 million) for Kroos, who had a year left on his contract in Bavaria.
Kroos, who sported the number eight jersey at his unveiling, denied that a disagreement with Bayern coach Pep Guardiola was behind his decision to leave the Bundesliga champions.
He said: "I had already been at Bayern for a long time and have won everything there, so I thought a new step would do me good."
The two clubs said the official transfer fee would remain secret, but the arrival of Kroos could hasten the departure from Madrid of his fellow German Sami Khedira, who now risks losing his midfield place and has been linked with Arsenal.
"A magnificent player has arrived, as he showed at the World Cup, where he was one of the best in the competition," said Madrid president Florentino Perez.
"His great performances in recent weeks do not come as a surprise, but as confirmation that we had chosen the right player."
Earlier, a Bayern statement confirmed they had agreed a deal with Real and that Kroos had signed a six-year contract.
"The two clubs have agreed to remain silent about the transfer fee," it said.
"We thank Toni Kroos for his time at Munich," said Bayern chief executive Karl-Heinz Rummenigge. "We have experienced great successes with him."
Kroos arrives at the Santiago Bernabeu with 11 trophies already on his CV, the World Cup the jewel in a collection including the 2013 Champions League, last season's Bundesliga title and the Club World Cup.
He joined Bayern as a teenager in 2007, making his first-team debut at the age of 17 and becoming a regular after an 18-month loan spell at Bundesliga rivals Bayer Leverkusen.
- Sports & Recreation
- Toni Kroos
- Bayern Munich
- Real Madrid