Media regulator Ofcom, which makes the appointment at the state-owned broadcaster, confirmed that Burns is joining the board immediately as chairman-designate.
His first task will be to appoint a new chief executive to replace Andy Duncan, who leaves next month.
Colette Bowe, Ofcom chairman, said: "I am delighted that we have been able to appoint someone of Terry Burns' calibre. He has an outstanding record of public service and a real depth of commercial expertise."
Burns had not been publicly linked with the chairman role up until now. Frontrunners for the job included Greg Dyke, former director-general of the BBC, and Richard Eyre, who was chairman of GCap Media.
When the search launched this time around, it was reported that Ofcom hoped to recruit a woman for the role but prospective candidates, including the National Lottery's Dianne Thompson, are thought to have turned down the opportunity.
Burns has previously had an input into the government's broadcasting industry policy. In 2005 he led the government's review of the BBC's governance, funding and remit in the run-up to its charter renewal.
His report recommended the then BBC board of governors be replaced by an external public service broadcasting commission with the power to direct licence fee funding to other public service broadcasters. However, the government instead decided on the less radical move of replacing the board with the BBC Trust.
Prior to being permanent secretary to the Treasury between 1991 and 1998, Burns was the Conservative government's chief economic adviser, a role he held for 11 years ending in 1991.
Burns is chairman of Abbey and has held non-executive directorships at companies including Legal & General and Pearson.