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."
Praising Meghan, who gave birth at 5:26am, Harry said: "How any woman does what they do is beyond comprehension, but we're both absolutely thrilled and so grateful to all the love and support from everybody out there. It's been amazing, so we just wanted to share this with everybody.
"I haven't been at many births. This is definitely my first birth – it was amazing, absolutely incredible. I'm so incredibly proud of my wife. And as every father and parent would say, your baby is absolutely amazing. But this little thing is absolutely to-die-for so I'm absolutely over the moon."
The couple announced the news via their Instagram account, Sussex Royal.
"We are pleased to announce that Their Royal Highnesses The Duke and Duchess of Sussex welcomed their firstborn child in the early morning on May 6th, 2019. Their Royal Highnesses’ son weighs 7lbs. 3oz," the statement reads.
"The Duchess and baby are both healthy and well, and the couple thank members of the public for their shared excitement and support during this very special time in their lives.
"More details will be shared in the forthcoming days."