The Category A-listed Victorian building is being upgraded to help it tell the stories of a town known around the world for the pattern that carries its name.
Renfrewshire Council has granted planning permission for the extension and external alterations plus listed building consent.
The museum is expected to draw audiences of 125,000 people a year - almost four times current numbers - from Scotland, the rest of the UK and abroad when it reopens in 2022.
In its response to the consultation, Historic Environment Scotland welcomed the proposal, which it described as “very positive, well justified”. It said that the plans “sensitively respond to the building and the requirement to provide step-free access”.
The Paisley Museum Reimagined scheme is the flagship project within Renfrewshire Council's £100m investment in cultural venues and outdoor spaces.
Councillor Marie McGurk, convener of Renfrewshire Council's communities, housing and planning board, said: “The Paisley Museum Reimagined project is at the heart of our bold plans to transform Paisley town centre in the years ahead - we are delighted it was able to pass this important milestone today.
"The designs which have been produced blend the old and the new - they will create a 21st-century visitor experience while preserving a key part of our fantastic architectural legacy, and ensure this much-loved building can stay at the heart of life in the area for generations to come.
"Paisley's name is already known around the world - and when the museum reopens in 2022 it will allow us to invite the world to come back to Paisley, providing new opportunities, life and footfall for the town centre and wider Renfrewshire area."
The project is expected to create a £79m boost for the local economy over 30 years, with 138 jobs supported during construction, and almost 50 jobs per year through revenue and visitor spending.