Burton graduated from London's Central Saint Martins college, also Alexander McQueen's alma mater, in 1997. She began working with McQueen in 1996.
"The creation of modern, beautifully crafted clothes was at the heart of Lee's vision. I intend to stay true to his legacy," Burton said in a statement.
She had earned raves for her completion of Alexander McQueen's final collection, which was presented in Paris in March.