Buckingham Palace said the boy, seventh in line to the throne, was born at 5.26am local time, weighing seven pounds and three ounces (3.26 kilograms).
Queen Elizabeth II, Prince Charles and Prince William were among the members of the royal family to have been informed and "are delighted with the news," it added.
Meghan's mother Doria Ragland is with her daughter at the royal couple's new Frogmore Cottage residence on the grounds of Windsor Castle west of London.
She was "overjoyed by the arrival of her first grandchild," Buckingham Palace added.
The siblings of Prince Harry's late mother Diana had also been told of the birth, it said.
Prince Harry was present for the birth of his first child and spoke candidly afterwards about the experience.
"How any woman can do what they do is beyond comprehension," he said.
"I am so incredibly proud of my wife and as every father and parent would ever say, your baby is absolutely amazing.
"This little thing is absolutely to die for. So I am just over the moon."
Well-wishes for the happy couple
Downing Street led the well-wishes as news of the royal birth broke on Monday afternoon in Britain.
"Congratulations to the Duke and Duchess of Sussex on the arrival of their baby boy," Prime Minister Theresa May said on Twitter.
The BT Tower in central London scrolled a message of congratulations.
Justin Welby, the Archbishop of Canterbury who married the couple last May, shared his joy on social media.
"May God bless the new family with love, health and happiness," he said.
In the US, the former first lady of the United States, Michelle Obama also sent her and Barack's congratulations.
Meghan Markle's former "Suits" co-stars also sent their best wishes on Twitter.
Prince Harry and Meghan revealed they were expecting their first baby in October, at the outset of a 16-day tour of Australia, New Zealand, Fiji and Tonga.
Commentators at the time believed the baby, the Queen's eighth great-grandchild, was due in late April.
The 34-year-old prince and Meghan, 37, reportedly opted for a home birth at Frogmore Cottage.
The move breaks with the tradition followed by Prince Harry's brother William and his wife Kate, whose three children were all born in the private wing of St Mary's Hospital in London.
Buckingham Palace also said last month Prince Harry and Meghan wanted to keep the arrival of their first child "private".
That also contrasted with William and Kate, whose newborns were immediately shown off by the couple in front of the world's media.
They have three children: five-year-old Prince George, Princess Charlotte, 4, and one-year-old Prince Louis.
Prince Harry said on Monday he expected to introduce the newborn "as a family" to the nation "in probably two days time".
He is set to visit the Netherlands on Thursday to mark the one-year countdown to the Invictus Games.
Britons have been on tenterhooks as April came to an end without news of the birth.
There has been fevered speculation in recent weeks over everything from the newborn's gender and name - to whether it will have Harry's ginger hair.
The prince said the couple were still thinking about names.
"The baby's a little bit overdue so we've had a little bit of time to think about it," he added.
"We're both absolutely thrilled and so grateful for all the love and support from everybody out there."
Names including Arthur, James and Albert are now hotly tipped by British bookmakers to be chosen by the couple.
Meanwhile, royal enthusiasts have been seen gathering in Windsor as news of the birth spread.
Meghan was a US actress before marrying Prince Harry last year at Windsor Castle.
She has been hailed as a breath of fresh air in a stuffy institution.
Meghan, who grew up in California and is a long-time advocate of holistic treatments, has reportedly raised eyebrows during her pregnancy with some of her new-age ideas and US glitz.
How she intends to bring up the baby will only increase the scrutiny.