The Germany international joined the Gunners from the Spanish giants on transfer deadline day on a "long-term contract".
"Dear fans of Real Madrid, Thanks for three wonderful years," he wrote on his official Facebook page. "My time at Real Madrid has been unique and special.
"Sometimes things unfold differently from what one would have expected a few days ago. Although I'm leaving Madrid I wish you and the team the best for the future.
"Now I'm looking forward to playing with my new team – The Gunners."
Ozil made a total of 159 appearances for Madrid in his three years with the former European champions, winning La Liga and the Copa del Rey in the process.