Mr Murray will step down later this month after four years in the role.
Mr Gillies, who joined Edrington as a non-executive director earlier this year, has had a number of prominent business roles during his career.
He remains a non-executive director of energy giant SSE and Barclays.
Mr Gillies also founded The Saltire Foundation, which helps develop entrepreneurial leadership in Scotland.
The announcement comes two weeks after Edrington reported a sharp rise in annual revenues and profit.
The Glasgow-based distiller's portfolio includes The Famous Grouse, The Macallan and Cutty Sark.
Mr Murray said: "I am very pleased to announce that Crawford will take over the chairmanship of Edrington and know that his extensive international business experience will be invaluable to the company and I wish him all the very best.
"It has been a privilege to work with so many great people who have driven commercial success and simultaneously promoted the values that set Edrington apart.
"I am confident that they will build on that success and wish them continued success in their future endeavours."
Former Scottish Enterprise chairman Crawford Gillies is to take over at the helm of spirits group Edrington, following the retirement of its chairman Norman Murray.