The fund is in addition to a £9.4m grant awarded to the project in January 2014.
Earlier this month, it was revealed that initial projected costs for the project had almost doubled, from £45m to £80m.
The project has since received additional funding from various sources, and on Tuesday, the HLF confirmed it too had increased its funding.
Carole Souter, Chief Executive of the Heritage Lottery Fund (HLF), said: "V&A Dundee will provide Scotland with a world-class museum of design while making a transformational contribution to the cultural, social and economic regeneration of the city.
"As a key member of the funding partnership, HLF's UK Board has agreed to commit an additional £4.5m budget to the project to help cover increased building costs. We can now look forward to seeing this iconic building start to take shape and realising its full potential."
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Welcoming the decision, Philip Long, Director of V&A Dundee, said: "This generous support from the Heritage Lottery Fund represents a significant boost to the new museum.
"Together with important recent decisions by Scottish Government and Dundee City Council it opens the door to an impending site start. We can now look forward to getting on and building this outstanding new museum, the first in the UK to be dedicated to design outside London.
"The renewed support of our funding and founding partners underlines the momentum and determination driving forward V&A Dundee, which will do so much to change the fortunes of this city. We very much appreciate these commitments and would like to thank to all involved."
Earlier this week, Dundee City Council agreed to allocate £6.5m of the authority's capital budget to the scheme, while talks are ongoing with the Scottish Government about developing a growth acceleration model. This would provide additional funding for the museum and the wider £1bn city waterfront regeneration project.
Work on the museum itself is scheduled to commence at the end of March, with a completion date of the end of 2017. A fit-out of the galleries and first exhibitions would follow, leading to the expected opening of the museum by June 2018.
(JP/MH)
