The duke and duchess's Instagram account @SussexRoyal also announced the birth saying: "It's a boy! Their Royal Highnesses the Duke and Duchess of Sussex are overjoyed to announce the birth of their child."
The most amazing experience
A beaming Harry said: "It's been the most amazing experience I could ever have possibly imagined.
"How any woman does what they do is beyond comprehension. But we're both absolutely thrilled.
"We're both absolutely thrilled and so grateful for all the love and support from everybody out there, it's been amazing, so we just wanted to share this with everybody."
Harry said the couple were "still thinking about names" for their son.
He said: "The baby was a little bit overdue so we've had a little bit of time to think about it.
"That's the next bit."
He said he planned to make another announcement in two days' time "so everyone can see the baby".
Harry added: "I haven't been at many births.
"This is definitely my first birth - it was amazing, absolutely incredible.
"I'm so incredibly proud of my wife.
"And as every father and parent would say, your baby is absolutely amazing.
"But this little thing is absolutely to-die-for so I'm absolutely over the moon."
What the latest odds say at the bookies for the royal baby's name
For a boy, Arthur, James, Philip, Edward and Albert are among the predicted choices.
William and Kate chose royal favourites for their children.
But Harry and LA-born Meghan could break the royal mould and take inspiration from American traditions, or perhaps the former actress will follow the celebrity trend of picking an unusual first name for her offspring.
Bookmakers were convinced that Harry and Meghan were expecting a daughter, but punters will now be betting that the couple will take inspiration from royal history when naming their son.
Arthur is one of the favourites at the bookmakers - as are Edward, James and Albert.
Other possibilities include a tribute to Harry's father, the Prince of Wales, with Charles, or the baby's great-grandfather, the Duke of Edinburgh, with Philip.
The duke could also pay his respects to his late mother Diana, Princess --of Wales, who died in a car crash when he was 12, by using Spencer - the princess's family name.
It is thought unlikely that Meghan will honour her estranged father, Thomas Markle.
But the LA-born former actress may decide to draw on American culture when picking a name.
The baby will only become a prince if the Queen steps in at the last minute by issuing a Letters Patent under the Great Seal of the Realm.
If this happens, the baby will become history's first Prince of Sussex.
The previous 19th century Duke of Sussex - an eccentric son of King George III - had both his marriages deemed illegal.
His mistress, Lady Augusta Murray, was not the Duchess of Sussex and his illegitimate son was plain Master Augustus Frederick d'Este.
Lady Augusta did, however, continually refer to herself as a duchess and a princess, and staff called her son Prince Augustus and her daughter Princess Augusta.
Baby Sussex will one day be entitled to become an HRH and a prince when his grandfather Charles eventually becomes King, but he or his parents could decide to keep his current title instead.
Live updates - follow our royal baby blog here for more as we get it