Carlo Ancelotti revealed his interest in the 21-year-old at the start of last month and the European champions have now completed the transfer, with the club confirming the news on Friday.
Silva becomes the second player to join Real in as many days after Martin Odegaard sealed his switch from Sromsgodset on Thursday.
Following months of speculation, the 16-year-old was officially confirmed as a Real player, describing the move as "a dream come true".
Despite the media storm around the transfer, Ancelotti has called for patience in expecting too much of Odegaard too soon.
"Martin is a very young and really quality player," he said. "We are very glad we have him as we really wanted to sign him.
"We trust he will have a great future here and what happens next is that we will take it easy.
"He will practice with the first team and then will play with Castilla - this will be coordinated with [Zinedine] Zidane. Everybody is happy he decided to come to Real.
"Odegaard symbolise the future of this team. He has a great future ahead and is a very important signing. It's great to have a young and skilled player."
