The practice - featured in the AJ's 40 Under 40 list - was appointed from a shortlist of seven to design the information resource centre and redesign the council's offices.
The practice saw off 6A Architects, Brady Mallalieu, Carey Jones, The Architects Practice and several others in the competition.
The Crafts Council is said to be making the resource centre the focus of the building, while improving public accessibility and increasing the headquarters' prominence.
The Crafts Council was set up in 1971 as an arm of Arts Council England and operates from the former Claremont Chapel, built by William Wallen in 1819.
The scheme is a major coup for Burd Haward, which is earning a reputation as one of the UK's hottest young practices.
The work will be phased to allow the Crafts Council to continue to operate from the building, with the first phase expected to start early in the New Year.