The 33-year-old made almost 500 appearances for the Blues following his arrival from Rennes 11 years ago, but last season he lost his place between the posts to Thibaut Courtois.
He has now moved across London to gain more regular starting action at the Emirates Stadium, where he has signed a "long-term contract".
"I'm really excited about joining Arsenal and can't wait to join up for pre-season. I have the same commitment to football, the same motivation and the same hunger for success as I had at the beginning of my career, and I love the challenges brought by the top quality players you face while playing in the Premier League," the 33-year-old told the club's official website
"When Arsene Wenger spoke to me about his ambitions for this club, and how he saw me as part of this team, the decision was clear."
It is believed that Arsenal have parted with around £11m to secure the Czech Republic international's signature.