The Heritage Lottery Fund will gift the money to redevelop the building into a museum, gallery, sport and cultural centre.
As revealed in the Evening Times in June, the Kelvin Hall will become the new museum for the Hunterian collections of Glasgow University, as well as a home for items from the Glasgow Museums collections.
The work will be done in two phases and the total cost is expected to be about £60m.
Colin McLean, head of the Heritage Lottery Fund in Scotland, said: "The Kelvin Hall is a hugely popular building, much loved by the people of Glasgow and beyond, so it is fitting the Lottery- playing public have had a hand in securing its future.
"This is a groundbreaking project that brings together civic, university and national heritage collections for the first time in the UK."
About 1.5 million objects, currently stored in various locations in the city, will be relocated to the new facility.
The building, opposite the Kelvingrove Art Gallery and Museum, will eventually become home to the Hunterian Collections Study Centre.
The work's first phase - transforming a large part of the building into a joint collections and study centre - is planned to be completed by the end of 2016. The second stage, creating new galleries and special exhibition and education spaces, is scheduled to end in 2020.
The museum and gallery will also house the only surviving complete suite of interiors by Charles Rennie Mackintosh.
The National Library Of Scotland's Scottish Screen Archive will also be given a new home, giving the public access to more than 100 years of history on film and video.
Kelvin Hall was built in 1926/27 to house large-scale exhibitions.
Archie Graham, chairman of Glasgow Life, which runs the city's museums, said: "This ambitious project will breathe new life into what is already a vibrant museums quarter, with Kelvingrove, the Hunterian and Riverside museums nearby."