Burleigh will join the group on 18 January. She has spent the past year as a retail marketing consultant at her own ACB agency.
She had previously been with Dixons for six years, rising to the post of marketing director – a role she held for almost three years.
Burleigh has also held marketing positions at Argos and PC World.
Donal Horgan, managing director for Musgrave GB, said: "Anna brings with her a wealth of retail experience and is an excellent addition to the GB Management team".
Musgrave was one of the brands to sign up to the government's sustainability pledge in March.