The 77-year-old, who was the third generation of his family to hold the position, feels the time is right to step aside.
Hill-Wood suffered a heart attack last year, and he told Arsenal's official website: "This has been a difficult decision as Arsenal Football Club has been a huge part of my life.
"Although I am feeling better I feel this is the right time for me to stand down. I would like to thank our majority owner, Stan Kroenke, for the respectful and dignified way he has handled this and I know Sir Chips is the right person to take the club forward."
The new chairman, 73, has been on the Arsenal board since 2005 and is a director of the Bank of England.
A statement on the club website said he would "work directly with Arsenal chief executive Ivan Gazidis on the club's football and business ambitions."
Keswick said: "I am sad to see Peter standing down due to his health and we all hope he continues well in his recovery.
"The Hill-Wood family has made an extraordinary contribution to Arsenal over many decades. They have been central to the Club's many achievements during this time.
"I am greatly honoured to have been appointed chairman of Arsenal Football Club. This is one of the great clubs in the game, recognised and loved by millions. I am looking forward to leading the club to future success."