emotional", while William said they were still choosing a name and joked that he already changed the baby's nappy.
The new third-in-line to the throne raised a tiny hand above his white blankets but remained peaceful, despite deafening cheers from well-wishers and the shouts of photographers outside St Mary's Hospital in London.
"It's a special time," said the 31-year-old Duchess of Cambridge, who was wearing a cornflower-blue dress with her brown hair flowing loose. "I think any new parent would know what this feeling feels like."
The duke, also 31, told reporters that "he's got her looks, thankfully" as Kate playfully demurred, then added: "He's got a good pair of lungs on him, that's for sure."
The royal baby's birth at 4:24pm (1524 GMT) on Monday sparked a global media frenzy, fuelled by the fascination that has surrounded the couple ever since they married in April 2011.
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 newborn prince became quite the buzz on Twitter too as many, including celebrities, took to Twitter to express how they felt about the newborn.
Kate initially held the baby as the royal couple emerged from the front door of the hospital's exclusive private Lindo Wing, before passing him to her husband, who was dressed in a blue shirt and jeans.
The scene was reminiscent of when William was introduced to the world by his parents Prince Charles and the late Princess Diana outside the same hospital wing in 1982.
Balding William joked that despite the baby boy's own thin coating of hair "he's got way more than me, thank God".
Asked about changing the nappies of his son, who weighed in at a healthy eight pounds six ounces (3.8 kilograms) when he was born, the prince said: "We've done that already."
"He's done his first nappy already," added Kate.
The traditional names George and James have been favoured by bookmakers as fit for a king, but the royal couple gave nothing away.
"We are still working on a name so we will have that as soon as we can," William said.
After speaking briefly to the press, the royal couple returned to the hospital before re-emerging minutes later with their son in a car seat.
Like millions of other fathers, William briefly appeared to wrestle with the car seat, before securing it in the back of a black Range Rover parked outside the hospital and driving his family back to their home at Kensington Palace.
Kate's sister Pippa was reportedly waiting for them at the palace.
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.
Cannon fire salutes rang out on Tuesday 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.
The couple had earlier received their first visitors when Kate's parents Michael and Carole Middleton, self-made millionaires who run a party supplies business, arrived in a humble black London taxi.
"He's absolutely beautiful. They're both doing very well and we're so thrilled," a beaming Carole Middleton said afterwards.
Charles arrived around two hours later with his second wife Camilla in a chauffeur-driven limousine.
"Marvellous, thank you very much, absolutely wonderful," said Charles.
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 featuring the Cliff Richard song "Congratulations."
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.