Robb left the umbrella body in November to serve as director-general of the Office of the Third Sector at the Cabinet Office.
Atkins, who takes up her position next week, has also worked at the NSPCC as head of policy and public affairs, and for the Labour Party.
"I'm looking forward to securing a more equal and effective partnership with government," she said.
Atkins is the latest in a series of senior appointments at the NCVO.
Richard Williams recently joined as director of enterprise, and Ben Kernighan, director of services and development at the NCVO, is now also deputy chief executive at the body.