The 33-year-old is the son of Arsenal majority shareholder Stan Kroenke, and he will combine his new role with his duties with the family's sporting empire in the United States.
"It is an honour to join the board of Arsenal," he said. "This reaffirms our family's long-term commitment to the club.
"Arsenal is in a strong position on and off the pitch. We are enjoying a period of growth for Arsenal and the Premier League.
"We look forward to an exciting future that includes winning trophies, increasing our already impressive support around the globe and extending our commercial opportunities.
"We strongly feel our experience in sports management, marketing and broadcast will be an asset to an already impressive board."
Announcing the appointment, Arsenal chairman Sir Chips Keswick said: "We are delighted to welcome Josh Kroenke to the board of Arsenal.
"He has great experience in running sports organisations and brings extensive knowledge of what is required to succeed as we develop our club for the future."