Ms Knight announced her departure from the role in April this year and is set to join Energy UK.
Mr Browne will take up his new appointment on 1 September.
He currently works as head of government relations at Morgan Stanley for Europe, Middle East and Africa. He has previously worked as mayoral policy director for economic development under Boris Johnson, responsible for economic and business policy for London.
Ms Knight said: “I am delighted that Anthony is to be the new chief executive of the BBA. He is well known to the staff and many of the members. We are really looking forward to his taking up the role.”
Mr Browne said: “It will be an honour to help lead the BBA, helping to resolve the complex issues still surrounding banking five years after the start of the financial crisis.
“In a time of change and uncertainty, the BBA has a critically important mission to ensure the banking sector returns to health, remains in Britain, and has the trust of the public and politicians.”