Councillors approved funding of up to £27.3 million at an Executive Committee meeting last week.
This new commitment, on top of a pledge of £5.3m, takes the local authorities project funding up to £33m, half of the estimated £66m cost of the scheme.
The Heritage Lottery Fund has pledged £15m towards the scheme, with the UK Government also committing £5m. A fundraising campaign is now underway to provide the remaining funds, with a target of £15m.
Work to transform the Grade A-listed building will see the creation of a dedicated space for special exhibitions and the conversion of offices into galleries. The roof of the building will be completely overhauled, while landscaped terraces and civic space will link the museum with surroundings of Pollok Country Park.
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In addition, over 90% of the 9,000-strong collection will be redisplayed with basement stores open to the public for the first time, and a new central core greatly increasing access to the many artworks.
The council's decision now means procurement can start for a main contractor. It is anticipated the refurbished museum will re-open in late 2020.
Depute Council Leader Archie Graham said: "The Burrell Collection is a masterpiece and the refurbished and redisplayed museum will be one of Scotland's finest cultural assets. Sir William's great legacy has been described as the greatest gift a city has ever received and we have a moral duty to ensure it is housed in the finest of buildings.
"The decision taken by the Council demonstrates our commitment to that great legacy and working with our partners, we will unlock the great potential of the Burrell Collection."
(LM/CD)
