The couple posed for their first picture together after the announcement on Monday.
Asked if his proposal was romantic, Harry replied: "Of course it was," before leaving with an arm around his fiancée who showed off her diamond engagement ring to photographers.
Harry, 33, Queen Elizabeth's grandson and fifth in line to the British throne, and Markle, 36, best known for her role in the US TV legal drama Suits, got engaged earlier this month in London.
"Prince Harry has informed Her Majesty The Queen and other close members of his family. Prince Harry has also sought and received the blessing of Ms Markle's parents," a statement issued by Prince Charles said on Monday.
The Queen, who had to give her assent for the union, and her husband Prince Philip are delighted, a Buckingham Palace spokesman said.
Harry and Markle, who is a divorcee, met in July 2016 after they were introduced through friends.
The prince, the youngest son of heir-to-the-throne Prince Charles and his first wife Princess Diana, publicly confirmed their relationship months later.
But it was not until September that they made their first public appearance together at the Invictus Games in Toronto.
"This is a time of huge celebration and excitement for two people in love and, on behalf of myself, the government and the country, I wish them great happiness for the future," British Prime Minister Theresa May said on Twitter.
The wedding is likely to attract huge attention across the world, as did the marriage of Harry's older brother William to Kate Middleton in 2011.
"We are very excited for Harry and Meghan," William and Kate said in a statement. "It has been wonderful getting to know Meghan and to see how happy she and Harry are together."
In his office's warning to the media, Harry referred to the sexism and racism directed at Markle, whose father is white and her mother African-American.
"We are incredibly happy for Meghan and Harry. Our daughter has always been a kind and loving person," Markle's parents Thomas Markle and Doria Ragland said in a statement.
"To see her union with Harry, who shares the same qualities, is a source of great joy for us as parents. We wish them a lifetime of happiness and are very excited for their future together."
Photo credit: Reuters. Caption: Meghan Markle shows of her diamond ring.