The work will result in improvements to the auditorium and other historic parts of the building. 20th-Century additions which are no longer fit for purpose will be replaced and new spaces created.
They will include a new foyer, rehearsal studios, an administration hub and improved catering facilities.
Access around the 1878 building will also be improved, with lifts to all levels and new viewing windows will highlight the Gorbals building's historic machinery and the auditorium.
The exterior will be given a new lease of life with statues of Shakespeare, Burns and the four Muses, originally on the roof, adorning the new facade.
Colin McLean, head of the Heritage Lottery Fund, said: "The Citizens Theatre is held in great affection in Glasgow and far beyond, although I don't think anyone would disagree it has seen better days.
"We have a shared vision of the Citz becoming a place where friends can gather for a coffee, enjoy outstanding performances and discover the wonderful working heritage behind its scenes."
The theatre has helped launch the careers of actors such as Stanley Baxter, Bill Paterson, Rupert Everett, Gary Oldman, Helen Baxendale, Alan Rickman and David Hyman.
Artistic director Dominic Hill said: "Our vision is to create a building which honours and celebrates the remarkable history of this theatre."