Mr Milne, whose parents were Scottish, rose through the ranks from a trainee to become controller of BBC Scotland before taking the top job in the corporation in 1981.
He was educated at Winchester College and New College in Oxford, and served as an officer in the British army regiment the Gordon Highlanders from 1949, before returning to study at Oxford.
He joined the BBC in 1954. During his time there, he produced shows such as That Was The Week That Was.
During his tenure in charge the BBC produced shows such as the coverage of Live Aid, launched EastEnders and introduced breakfast TV.
Mr Milne was a fierce defender of the licence fee, which was under pressure from the Government.
He famously told a committee of MPs that it represented the "best bargain in Britain".
He was eventually forced out in 1987 after a series of rows with the Thatcher government over the impartiality of the BBC.
Mr Milne, who was the oldest living former director-general, had suffered a series of strokes and passed away on Tuesday, according to his family.
Acting director-general Tim Davie said: "He was a charismatic editorial figure ... leaving a raft of outstanding programmes, as well as defending editorial independence during a turbulent political period."
BBC Scotland Director Ken MacQuarrie said: "Both broadcasting and Scotland have a lot to thank of Alasdair. He was a fearless leader and a stout defender of the independence of the BBC.
"He was also instrumental in helping to establish Gaelic children's programmes and was a keen supporter of Scotland's cultural life, particularly traditional music. Scotland is a poorer country without him."