Introducing him to the cameras earlier in the day, Meghan said: "He has the sweetest temperament, he's really calm. He's been the dream."
As they laughed, Prince Harry said: "I don't know who he gets that from."
The announcement came after the Queen and Duke of Edinburgh met their eighth and newest great-grandchild for the first time at Windsor Castle.
By deciding to call their son Archie Harrison Mountbatten-Windsor, Harry and Meghan have chosen not to use a title for their first born.
As the first born son of a duke, Archie could have become Earl of Dumbarton - one of Harry's subsidiary titles - or have been Lord Archie Mountbatten-Windsor, but instead he will simply be Master Archie Mountbatten-Windsor.
BBC royal correspondent Jonny Dymond said it was a "strong indication that they don't want to bring him up as a formal 'royal'".
Talking about their first few days as parents, Meghan said: "It's magic, it's pretty amazing. I have the two best guys in the world so I'm really happy."
The duke added: "It's great. Parenting is amazing.
"It's only been two and a half days, three days, but we're just so thrilled to have our own little bundle of joy."
On which parent the baby resembled more, Harry said: "Everyone says that babies change so much over two weeks.
"We're basically monitoring how the changing process happens over this next month really. But his looks are changing every single day, so who knows."
Later, asked to show more of their son's face to the cameras, Meghan laughed as Harry joked: "He's already got a little bit of facial hair as well, wonderful."
Meghan added: "Thank you everybody for all the well-wishes and kindness, it just means so much."