The building, based near Waverley railway station, will bring together around 3,000 civil servants from a range of UK Government departments who are due to move in from spring next year.
The keys to the building were formally handed over last month.
Secretary of State for Scotland David Mundell said: "I am absolutely delighted that Her Majesty has agreed to the naming of the new UK Government Hub in Edinburgh as Queen Elizabeth House.
"The royal title is hugely fitting, given the hub will be the focus of the UK Government's work in Scotland, hosting 10 UK Government departments.
"The building will contain a dedicated Cabinet room, the first of its kind outside of London, and I very much look forward to inviting the Cabinet to meet in the building once it is open for business.
"I am very grateful to Her Majesty for acknowledging the importance of the building by conferring the royal title."
As well as the Secretary of State's Office, the hub will also house the Competition and Markets Authority, the Office of the Advocate General for Scotland, the Office for Statistics Regulation, the Information Commissioner's Office, the Health and Safety Executive, the Office for Statistics Regulation, the Government Actuary's Department, the Cabinet Office and HM Treasury.
Work is also under way on building the first Glasgow hub for the UK Government.