In a statement released through Buckingham Palace, the monarch said the Royal Family agreed “there will be a period of transition in which the Sussexes will spend time in Canada and the UK”.
She added the family "respects and understands" the couple's "wish to live a more independent life as a family while remaining a valued part of my family."
The Queen's statement in full
The statement from the Queen said: "Today my family had very constructive discussions on the future of my grandson and his family.
"My family and I are entirely supportive of Harry and Meghan’s desire to create a new life as a young family.
"Although we would have preferred them to remain full-time working Members of the Royal Family, we respect and understand their wish to live a more independent life as a family while remaining a valued part of my family.
"Harry and Meghan have made clear that they do not want to be reliant on public funds in their new lives.
"These are complex matters for my family to resolve, and there is some more work to be done, but I have asked for final decisions to be reached in the coming days."
ITV News Royal Editor Chris Ship on the unprecedented statement from Buckingham Palace - and what could happen next in the Harry and Meghan saga
The Duchess flew to Canada a few days ago where the family spent an extended festive break in the province of British Columbia.
The royal crisis engulfed the Royal Family last week and made headlines around the world, after Prince Harry and Meghan announced they wanted to split their time between the UK and North America and become "financially independent."
The Queen was said to have been left the Queen feeling “hurt” by the couple's bombshell announcement.
It is thought William travelled to the estate in Norfolk from his Kensington Palace apartment and Harry from Frogmore Cottage near Windsor Castle.
Charles was in Oman to attend a condolence ceremony following the death of Sultan Qaboos bin Said al Said, and travelled back to the UK on Sunday evening.
The Queen attended church at Sandringham on Sunday where wellwishers expressed sympathy for her, and clapped as she left, with some members of the public saying that Harry and Meghan should not receive any more taxpayers’ money.
The 93-year-old monarch was pictured arriving for the service wearing a hearing aid for what is believed to be the first time in public.
Harry and Meghan’s statement – and a new official website – have thrown up important questions about funding for the couple’s round-the-clock security, media access to their royal events and how they will pay for their future lifestyles.
Other important issues to resolve include who will pay for the substantial security cost of protecting the couple and their baby son if they spend large portions of the year in Canada and possibly America.
Canadian prime minister Justin Trudeau said there have been "no discussions" about the details of the Duke and Duchess of Sussex's move to the country.
Speaking to Global News on Monday, Mr Trudeau said there had been no talks on security and the impact on taxpayers in his country.
He added: "There will be many discussions to come on how that works.
"But there's still a lot of decisions to be taken by the Royal Family and by the Sussex's themselves as to what level of engagement they choose to have and these are things that we are supportive of."
ITV News Correspondent Rachel Younger says Canadians are more interested in the weather than the royals right now
On Thursday, the Queen, Charles and William gave orders for their households to work with the Sussexes’ team to quickly find a “workable solution” to their desire to change the direction of their royal lives, but still support the Queen.
The 93-year-old monarch wishes for the crisis to be resolved "at pace."
Some at Buckingham Palace are reported to fear that Harry and Meghan could conduct a potentially damaging television interview if they do not get what they want.
In an article for The Sunday Times, broadcaster Tom Bradby – a friend of both Harry and William – said: “I have some idea of what might be aired in a full, no-holds-barred, sit-down interview and I don’t think it would be pretty.”
Watch the latest episode of our series The Royal Rota: Royal Editor Chris Ship is at Sandringham on the global reaction to Harry and Meghan's decision to step down