Farnsworth, who is currently project executive for BBC London 2012, will lead the newly created "visual journalism unit", which will combine the existing TV and online graphics departments into one multiplatform team.
The new unit, which will be based in the multimedia newsroom at New Broadcasting House, W1, will provide "graphics, data visuals and editorial content across TV, connected TV, PCs, tablets and mobiles for UK and global audiences", the BBC said in a release.
The role was advertised internally and not open to external candidates.
Farnsworth, a former editor of the BBC's One and Six O'Clock News, deputy editor of Newsnight and bureau chief in Washington DC, said in a statement that she is "determined to put our work at the heart of the BBC's journalism".
She will take up her role after the Olympics and Paralympics.
Mary Hockaday, head of the BBC newsroom, who talked about the new role at the Global Editors Network conference in Paris last month, added: "This is an important new role for BBC news, to allow us to build on our strong record in TV graphics, design and digital visual journalism.
"We have a great opportunity at New Broadcasting House for further transformation and innovation in an increasingly important aspect of story-telling across a range of platforms."