Dr Glover will take up the post, and the associated Professorship of Plant Systematics and Evolution to which she has been elected, in July 2013.
Dr Tim Upson, Curator at the Garden, has been acting director since January 2011 following the retirement of Professor John Parker.
Dr Glover, currently Reader in Evolution and Development in the Department of Plant Sciences, said: "The Botanic Garden is a central and much-loved part of both the University and the wider community. It is a great privilege and honour to be asked to lead its continued development.
"I am very much looking forward to working with the Garden's highly-skilled and dedicated staff to develop further the collections and to ensure they play their full part in botanical research and teaching, both in the University and worldwide."
Dr Glover read Plant and Environmental Biology at St Andrews before completing her PhD at the John Innes Centre in the molecular genetics of cellular differentiation in the plant epidermis.
She came to Cambridge first as a Junior Research Fellow at Queens' College, before progressing from Lecturer to Reader in the Department of Plant Sciences. A Fellow of the Linnean Society, she was awarded the Linnean Society Bicentennial Medal in 2010 and she received the William Bate Hardy prize from the Cambridge Philosophical Society in 2011.
The Botanic Garden in Cambridge was established on its current site by Professor John Stevens Henslow, mentor to Charles Darwin, and opened to the public in 1846. The Garden curates, develops and makes accessible to the public the living plant research collection of the University.
Professor Keith Richards, Chair of the Botanic Garden Syndicate, said: "We are delighted that Beverley will be the new Director of the University's Botanic Garden. She is well-known to everyone in the Garden, having served on the Syndicate for ten years, and has already made many highly-valued contributions to its outreach programme and to its integration into University teaching.