Hill-Wood, who has been Arsenal chairman since 1982, is recovering after a heart attack last year and stood down for health reasons.
Keswick, 73, has been on the Arsenal board since 2005. He is the former chairman of Hambros Bank and a director of the Bank of England. In his role as chairman, Keswick will work directly with Arsenal chief executive Ivan Gazidis on the club's football and business ambitions.
The appointment breaks an Arsenal dynasty going back to 1927 as Mr Hill-Wood's father Denis and grandfather Sir Samuel were Arsenal chairmen before him.
Arsenal majority owner Stan Kroenke said: "Peter has been a wonderful custodian of this club.
"He has been a member of the Arsenal board for 51 years and his vision and steadfastness have been central to the successes of this great club over that time.
"Peter will be sorely missed but we will be ensuring we recognise his many contributions to the club in an appropriate way."
On the new appointment, Kroenke added: "I am delighted that Sir Chips has agreed to become chairman.
"He has a well-respected track record in business and he will provide further direction and support in our ambition to compete at the top of the game here and in Europe and to win trophies."