The Duchess of Cambridge told the massed international media that it had been a "very emotional" experience, while the duke said that they were still working on a name for the third-in-line to the throne.
The baby raised a tiny hand above his white blankets but remained quiet and peaceful, despite cheers from well-wishers and shouts of hundreds of photographers outside the St Mary's Hospital in London.
The baby was born at 4:24pm (1524 GMT) on Monday after at least 10 hours of labour, weighing a healthy eight pounds six ounces (3.8 kilograms), ending weeks of anticipation around the globe.
Kate, wearing a loose-fitting cornflower-blue dress, held the baby as the royal couple emerged from the front door of the hospital's private Lindo Wing, before passing him to her husband.
"He's got a good pair of lungs on him, that's for sure," 31-year-old William - the son of Prince Charles and the late Princess Diana - told the press.
"It's a special time," added Kate, also 31. "I think any new parent would know what this feeling feels like."
After speaking briefly to the press, William and Kate returned to the hospital before re-emerging minutes later with their son in a car seat.
William secured the seat in the back of a black Range Rover parked outside the hospital, before driving his family back to their home at Kensington Palace.
Kate's sister Pippa was reportedly waiting for them at the palace.
The couple had earlier received their first visitors with heir to the throne Prince Charles and Kate's parents Michael and Carole Middleton - all first-time grandparents - turning up at the hospital.
Congratulations have poured in from around the globe for the baby, a great-grandchild of Queen Elizabeth II, who will one day reign over Britain and 15 other Commonwealth realms around the world.
The baby remains officially nameless, although bookmakers have picked George and James - traditional names that hark back to previous kings - as favourites.
Cannon fire salutes rang out at the Tower of London and Green Park in celebration of the birth, while the bells at the 11th century Westminster Abbey pealed across the capital for three hours.
By mid-afternoon, Kate was well enough for a visit from her parents, self-made millionaires who run a party supplies business. They arrived in a humble black London taxi and initially looked bewildered by the sheer scale of the international media presence.
"He's absolutely beautiful. They're both doing very well and we're so thrilled," a beaming Carole Middleton told the media when she had recovered her bearings.
Charles arrived around two hours later with his wife Camilla in a chauffeur-driven limousine.
"Marvellous, thank you very much, absolutely wonderful," said Charles, who earlier in the day had been on a tour of Yorkshire in northern England.
Hordes of journalists had camped outside the hospital for three weeks waiting for the baby, testament to the enduring appeal of the British monarchy and particularly the glamorous William and Kate.
At Buckingham Palace, crowds straining for a glimpse of the official birth announcement on a gold easel in the forecourt were treated to a special edition of the Changing the Guard ceremony.
The Queen's Guards, resplendent in red tunics and bearskin hats, performed Cliff Richard's "Congratulations", to cheers from well-wishers and tourists outside the gates.
The baby will be titled His Royal Highness, Prince (name) of Cambridge -- the blank to be filled in when his name is announced.
William's name was not announced for a week, while the world had to wait a whole month when Charles was born in 1948.
William and Kate did not know the sex of their child until he was born, although the duchess reportedly told a soldier at a St Patrick's Day parade in March: "I'd like to have a boy and William would like a girl."
It is the first time since 1894 that three direct heirs to the throne have been alive at the same time, and the 87-year-old queen said she was "delighted" at the birth of her third great-grandchild.
William and Kate are hugely popular and have been widely credited with revitalising the British royals following decades of scandal and the death of William's mother Diana in a car crash in 1997.
More than 25,300 tweets a minute were sent immediately after news of the birth broke on Monday night, Twitter said, while the hashtag #RoyalBaby was used 900,000 times in the first 24 hours after Kate went into labour.
US President Barack Obama led the international messages of congratulations, which also poured in from Australia, New Zealand, Canada, France, Israel, Japan and Singapore.
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@Pashtun: People with compassion, love, open thought, those who value what all religions teach......... they care because careing does does not diminish them, it strengthens them.
who cares....