Richard Hunter is leaving after 33 years with the spirits company including two decades in the finance function at the maker of The Macallan, Cutty Sark and Famous Grouse.
It marks a return to the whisky industry for Mr Short, currently finance director at AG Barr, who previously worked for William Grant & Sons for 12 years.
He is expected to take up his new role towards the end of August.
Soft drinks maker AG Barr confirmed the company had started searching for a replacement for Mr Short.
Chairman Ronnie Hanna said: "Alex has been an important member of our management team and board during his time with us.
"We would like to thank him for his significant contribution in that time and wish him well in his move back into the spirits industry with Edrington."
Ian Curle, Edrington chief executive, paid tribute to Mr Hunter and said he was looking forward to working with Mr Short. He added: "Richard has been instrumental in the growth of Edrington, and his career has spanned the transformative integration of Highland Distillers and Brugal.
"I am delighted that Alex is joining Edrington - he brings a wealth of experience of financial directorship as well as knowledge of the drinks industry from his time at AG Barr and William Grant & Sons.
"He will be an important member of our board and I look forward to working with him."
Mr Short is also a non-executive director at Goals Soccer Centres.