The wedding will take place in Spring 2018 with further details about the wedding day to be "announced in due course".
Harry, 33, currently fifth-in-line to the British throne, and Markle, 36, best known for her role in the US TV legal drama Suits, became engaged earlier this month, Clarence House said.
"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," the statement said.
Queen Elizabeth II and her husband, Prince Philip, "are delighted" with the news, Buckingham Palace said.
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Harry's older brother, Prince William, and his wife, the Duchess of Cambridge, said they were "very excited for Harry and Meghan. It has been wonderful getting to know Meghan and to see how happy she and Harry are together."
Markle's parents Thomas Markle and Doria Ragland said in a statement: "We are incredibly happy for Meghan and Harry. Our daughter has always been a kind and loving person. 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."
Kensington Palace said Harry and his new fiancee would be appearing together later on Monday local time.
The palace tweeted: "Prince Harry and Ms Meghan Markle will appear for a photo call at Kensington Palace later this afternoon. Stay tuned!"
The couple will also record a broadcast interview to be aired on Monday evening in the UK.
Speculation that the pair were about to announce their engagement intensified last week, with television crews turning up outside Kensington Palace and Buckingham Palace.
With Markle reportedly putting plans in place to move her two beloved dogs - Guy and Bogart - to the UK, and landing in London to be reunited with Harry, an announcement was seen as imminent.
The confirmation has come a week on from the Queen and the Duke of Edinburgh's platinum wedding anniversary.
The couple met in July 2016 after they were introduced through friends.
The prince publicly confirmed their relationship months later in a rebuke to the media, which had been intruding into Markle's private life, but it was not until September that they made their first public appearance together at the Invictus Games in Toronto, a sports event for wounded veterans.
The couple will live in Nottingham Cottage at Kensington Palace.
