Speaking to the media, the proud new dad said both Meghan and the 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. The couple are expected to step out publicly with their baby in two days time.