The extensive revamp will see the A-listed Burrell museum closed for three years, and the institution, which opened in 1983, will receive a new roof, new glazing, and a major internal development which will open up two extra floors to the public.
It is expected that the museum will close in early 2016 and reopen in 2019.
When it re-opens the museum will be able to display more than 90 per cent of the 9,000-strong collection collected by Sir William Burrell and gifted to the city in 1944.
The basement stores will be opened to the public for the first time and there will also be improved cafe and shop with a new “civic events space” outside.
The building which houses the Burrell Collection is in urgent need of refurbishment – galleries have been closed because of danger of damage to objects and paintings as a result of water ingress, particularly through the roof.
The new master plan for the revamp has produced the first detailed estimated costs for the project, with the figure expected to be in the region of £60m to £66m.
The council will fund up to 50 per cent of the total cost, with the remainder split between the HLF, a fundraising campaign and grants from other public bodies.
Heritage grant applications are assessed in two rounds.
A first-round pass is given when HLF has endorsed outline proposals and earmarked funding, and may also include an immediate award to fund the development of the project.
Detailed proposals are then considered by HLF at the second-round and, as long as they have progressed to the funds’s satisfaction, the money for the project is confirmed.
The Burrell Collection’s £15m grant is the biggest amid a £100m series of funds announced by the HLF today.
Colin McLean, head of the HLF in Scotland, said: “The Burrell Collection was one of the greatest gifts ever given to a city and the seed that helped Glasgow grow into the thriving cultural centre it is today.
“Thanks to National Lottery players, we are delighted to support plans to breathe new life into it so that it can continue to be a place of joy and inspiration for local, national and international visitors for many years to come.”
Sir Angus Grossart, the chairman of Burrell Renaissance, said: “We have been working hard to liberate the great potential of the Burrell Collection and for Sir William’s great gift to be placed on a global stage.
“It is key that we provide a home worthy of these incredible treasures and their influence.
“We deeply appreciate the decision by the Heritage Lottery Fund to support this project which is a ringing endorsement of our plans.
“It is yet another, very positive, step as we carry forward the torch handed to us by Sir William and seek to place his extraordinary collection within the international context which it deserves.”
Archie Graham, the chair of Glasgow Life, said: “The Burrell Collection is of international importance and the undoubted jewel in our cultural crown.
“We are delighted that the Heritage Lottery Fund has fully backed our ambitious plans for a full refurbishment and redisplay, which will massively increase access to some of the treasures which Sir William devoted his life in collecting.”