The architecture critic and writer is the surprise choice to take over the post following the controversial departure of Alice Rawsthorne earlier this year.
Sudjic said: 'This is the best possible job for someone who is passionate about design.
'[It] is a once-in-a-generation chance to create an exciting new museum for London that the world will come to and enjoy.'
He will take the reins at the beginning of September.
Luqman Arnold, chairman of the museum's trustees, said: 'Shaping the future and vision of the Design Museum and delivering what will be an ambitious project requires someone with an exceptional range of experience, expertise and authority.
'Deyan has an exceptional track record as a curator, an educator and a fund-raiser, and as a leader and deliverer of complex projects.'