Real had earlier confirmed a deal had been agreed but neither club have given financial details. Spanish and Italian media reported the transfer was worth 15 million euros plus 3 million linked to future achievements.
"I think Inter is an amazing club, one of the biggest clubs in the world, so I'm very happy to be a player of Inter. It has been long, but finally I am here," Sneijder told a news conference after being given the number 10 shirt.
"I can say that Mourinho is 100 percent behind me so this was a very big part of my decision. I think I'm an attacking midfielder, so playing behind the strikers, yes, it's my position."
Mourinho said on Thursday the 25-year-old could feature in Saturday's city derby with AC Milan but then decided to omit Sneijder from his initial match squad.
The Portuguese had again repeated his desire for a playmaker after Inter's unimaginative 1-1 draw at home to Bari in Sunday's Serie A opener.
Inter, who failed to sign Chelsea's Frank Lampard last year and missed out on team mate Deco in this transfer window, often lacked creativity last season despite easily sealing a fourth straight Serie A title.
Sneijder said on Thursday that Real had treated him shabbily, having initially indicated the former Ajax player was part of new coach Manuel Pellegrini's plans for the new season.
If Mourinho changes his mind over Saturday's squad, Sneijder could face AC Milan striker Klaas Jan Huntelaar, another Dutchman deemed surplus to requirements at the Bernabeu following their unprecedented spending spree which included the signings of Cristiano Ronaldo and Kaka.
Real's Dutch winger Arjen Robben has also joined Bayern Munich.
