Speaking to the media from outside Frogmore cottage, the proud new dad said both Meghan and their baby are doing "incredibly well" and he is "very proud" of his wife.
He also revealed that the couple are yet to decide on a name for their newborn, who weighed in at 3.2kgs.
"It's been the most amazing experience I could ever have possibly imagined," Harry continued.
"We're both absolutely thrilled."
It is understood Meghan went into labour at 1am on Monday morning. Prince Harry and her mother, Doria Ragland, were by her side during the birth.
The new parents are expected to step out publicly with their baby in a few days after they've had time to celebrate the exciting news as a family.