Prince Harry said they were "absolutely thrilled" and thanked the public for their support during the pregnancy.
He said Meghan and the baby were doing "incredibly well", adding that they were still thinking about names for the infant, who was delivered at 05:26 BST.
Buckingham Palace said the baby weighed 7lbs 3oz (3.2kg), and that the duke was present for the birth.
It added: "The Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall, the Duke and Duchess of Cambridge, Lady Jane Fellowes, Lady Sarah McCorquodale and Earl Spencer have been informed and are delighted with the news.
"The duchess's mother, Doria Ragland, who is overjoyed by the arrival of her first grandchild, is with their Royal Highnesses at Frogmore Cottage."
The baby boy is seventh in line to the throne, behind the Prince of Wales, the Duke of Cambridge and his children - Prince George, Princess Charlotte and Prince Louis - and Prince Harry.
Speaking after the birth, Harry added the child had been "a little overdue" and that he planned to make another announcement in two days' time "so everyone can see the baby".
Asked what it was like to be present for the birth, he laughed and said: "I haven't been at many births. This is definitely my first birth. It was amazing, absolutely incredible, and, as I said, I'm so incredibly proud of my wife.
"As every father and parent will ever say, you know, your baby is absolutely amazing, but this little thing is absolutely to-die-for, so I'm just over the moon."
He added: "It's been the most amazing experience I could ever have possibly imagined.
"How any woman does what they do is beyond comprehension."
The baby is a dual citizen, having automatically acquired US citizenship, via Meghan, at birth.
Speaking to the Sun, Meghan's father, Thomas Markle, congratulated the new parents and spoke of his pride at the birth.
"I am proud that my new grandson is born into the British royal family and I am sure that he will grow up to serve the crown and the people of Britain with grace, dignity, and honour," he said from his home in Mexico.
Charles Spencer - brother of Diana, Princess of Wales and uncle to Harry - tweeted his congratulations.
He wrote: "Really very lovely news today - many, many, congratulations!" (BBC)