The Heritage Lottery Fund announced the support for the planned waterfront museum, which is scheduled for completion in 2016.
An international competition to design the landmark building was won by Japanese architect Kengo Kuma.
The lottery grant will go towards the capital costs of the building, including the development of galleries showcasing Scotland's design heritage.
Philip Long, director of the V&A at Dundee, said the award was a "massive vote of confidence" in the project.
He said: "Scotland's achievement in design is world-class and centuries old. The development of V&A at Dundee will for the first time provide Scotland with a dedicated centre to celebrate that achievement, as well as a place to see and understand the design creativity of other countries and cultures.
"We are now looking forward to being on site and getting on with our pre-opening programme across the country."
Dame Seona Reid, deputy chair of the Heritage Lottery Fund, said: "Today's decision brings HLF's investment in Scottish museums and art galleries to over £177 million.
"This new world-class museum is an exciting addition to the sector providing a source of education and delight for visitors, while making a transformational contribution to the cultural, social and economic regeneration of the city."
The funding was also welcomed by Culture Secretary Fiona Hyslop and Scottish Secretary Alistair Carmichael.
The Scottish Government has committed £15 million to the £45 million project and Creative Scotland is currently considering a funding submission. Museum directors are seeking a further £15 million from private sources, with £6.7 million already raised.
The Scottish Design Galleries within the V&A will tell the story of Scotland's contribution to global design, with exhibits ranging from the seventeenth century to the present day.