The midfielder signed a six-year deal at the 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."
The Spain Under-21 international was delighted to be catching the eye of the Liga giants and revealed that Blancos star Xabi Alonso convinced him to make the move to the capital-based club this summer.
"I went home to my town after the European Under-21; my agents called me and told me that Real Madrid was interested," Illarramendi added.
"I was flattered because having a club as important as Real Madrid interested in you doesn’t happen every day. I talked to the president, who was kind to me and I appreciate that.
"I also spoke to Xabi Alonso, who advised me to come because he said it was incredible. He's a great player, he’s always been a leader and I hope to learn a lot playing with him."
The midfielder believes there is team on the planet better than his new club and thanked his new supporters and bosses for the welcome he received on Saturday.
"It’s clear that Real Madrid are the best club in the world; they’ve proved it over the years," the Spaniard told the club's official website. "The president and his entire board of directors made a massive effort so that I can be here today and there’s nothing more I can do than thank them as well as all the fans who gave me such an incredible welcome.
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 their European Championship-winning campaign earlier this summer and has been likened to team-mate Xabi Alonso for his passing-based game.