The midfielder signed a six-year deal at Santiago Bernabeu on Friday after Madrid met the sizeable release clause in his contract, with the player joining from Real Sociedad.
"Today is a very special day for me," he said. "Getting the chance to play here is a very unique opportunity. I will give everything for this crest.
"I would like to show my appreciation to Real Sociedad and their fans for everything they've done for me."
Los Blancos president Florentino Perez also credited Sociedad, and said Illarramendi would be well supported by the Madrid faithful.
"We are happy to welcome Asier Illarramendi to Real Madrid," Perez said. "I would like to show my appreciation to Real Sociedad and their president for making this possible.
"The fans will support you in the good and bad times."
Illarramendi starred for Spain's Under-21 side in its European Championship-winning campaign earlier this summer and has been likened to teammate Xabi Alonso for his passing-based game.