The Duke and Duchess of Sussex have named their newborn son Archie Harrison Mountbatten-Windsor.
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."
Archie Harrison: The meaning behind the name
A surprise choice, Archie was not among the bookmakers' favourites of Alexander, Arthur and Albert.
"I don't think anyone of us saw either of these names coming," says Joe Little, managing editor of Majesty royal magazine.
As far as he is aware, Archie does not have any British royal connotations - and Harrison too is a totally new name for the Royal Family.
Archie means "genuine", "bold" or "brave" - and is more popular in Britain than the US. It was originally a shortened form of Archibald but is now often used as a name on its own.
Pictures of the new family's first photocall were then shared on the SussexRoyal Instagram account.
"The Duke and Duchess of Sussex are delighted to share their first public moment as a family," the post said.
The images shared on social media were taken by Chris Allerton, who was the couple's private wedding party photographer, and took the picture of the newlyweds watching fireworks in the grounds of Frogmore House at their evening reception.
The infant, who is seventh in line to the throne, was delivered at 05:26 BST on Monday, with Harry saying after the birth that he and Meghan were "absolutely thrilled".
The Duke of Cambridge has welcomed his brother to "the sleep deprivation society that is parenting".
Prince William - a father-of-three - said he looked forward to seeing the new parents "when things have quietened down".
The Prince of Wales has also expressed his joy at the birth.
During an official visit to Germany with the Duchess of Cornwall, Prince Charles said: "We couldn't be more delighted at the news and we're looking forward to meeting the baby when we return."
A statement added that Meghan's mother, Doria Ragland, was "overjoyed at the arrival of her first grandchild" and was with her daughter at Frogmore Cottage - the new parents' home on the Windsor Estate.
By BBC News correspondent Helena Wilkinson
As settings for first family photographs go this was rather more grand than most, to say the least - St. George's Hall in Windsor Castle is a room used for state banquets.
But for Meghan and Harry it is a special room already filled with memories - it was where they held their wedding reception last year. Today they created new ones.
Wrapped in a white shawl and hat, baby Sussex was oblivious to it all, sleeping soundly in his father's arms.
As soon as the pictures of the newborn emerged news crews outside the castle beamed them live across the world.
This two-day baby may only be seventh in line to the throne but interest in him is huge.
Something his parents, who guard their privacy closely, will be very aware of.
Buckingham Palace said Harry was present for the birth, which is understood to have taken place in hospital as opposed to at home.
The baby will not automatically become a prince - although he could get that title if the Queen steps in.
Harry and Meghan might otherwise want him to inherit one of Harry's titles, such as Earl of Dumbarton, or they could reject any royal title at all.
The baby weighed 7lbs 3oz (3.2kg), making him lighter than each of his cousins.
Prince George was 8lb 6oz (3.8kg) when he was born in July 2013. His sister Princess Charlotte was 8lbs 3oz (3.7kg) when she arrived in May 2015, while their younger brother Prince Louis weighed 8lb 7oz (3.8kg) when he was born in April last year.
The average weight of a baby in the UK is about 7lb 7oz (3.4kg).