The Labour politician – an expert of 18th- and 19th-century history – will leave his post as MP for Stoke-on-Trent Central to lead the world's largest museum of decorative arts and design.
He is expected to take up the post in the coming months, picking up from former director Roth, who left the role at the end of 2016 to return to his native Germany. Roth's move was rumoured to have been fuelled by the Brexit vote.
Hunt, who was previously the shadow secretary of state and the shadow minister for education, is a specialist on Victorian urban history. He has authored several books, is a regular television history broadcaster, and lectures at Queen Mary University of London.
He is also a fellow of the Royal Historical Society, a founder of the Stoke-on-Trent Literary Festival and a patron of the British Ceramics Biennial.
"He has a highly compelling mixture of experience across public life, the arts, history, education and academia, and knows our collections well from his writing and broadcasting," said V&A chairman Nicholas Coleridge.
"In addition, he is an informed and articulate leader and communicator on numerous facets of culture, both historic and contemporary, and I greatly look forward to working with him at the V&A."
Hunt said he was "delighted and honoured" to take up the position, just as the museum is about to open a major new extension by architect Amanda Levete.
"I have loved the V&A since I was a boy, and today it is a global leader in its unrivalled collections, special exhibitions, academic research and visitor experience," he said.
"It is a moment of transformation and renewal for the V&A, with the upcoming opening of the new Exhibition Road entrance, and new sites and galleries in Dundee, China and Stratford," he continued.
"The combination of the power of the collections and expertise of an inspirational team is what makes the V&A the world’s greatest Museum of art, design and performance."