Kelly, 44, who was born and studied in the US before moving to Korea as a political science professor, focuses entirely on the camera as he attempts to blindly hand off his daughter, who is clearly curious as to who he is talking to.
And his parental problems soon double as a baby also excitedly makes his way into the room under his own power in a walker.
To complete the farce, his wife Jung-a Kim then comes skidding through the threshold.
She grabs the two youngsters and attempts to drag them out of the door, but one of them can be heard wailing and the baby’s walker suddenly won’t fit back through the door.
Eventually, she manages to get them both out, and the interview continues.
When the interview finishes, broadcaster James Mernendez says: ‘There’s a first time for everything. I think you’ve got some children who need you!’
And after the segment had finished, the presenter admitted on Twitter that he had struggled to keep it together.
He posted a link to the video, with the words: ‘Hard to keep a straight face.’
Afterwards, he added: ‘It was the desperate reach for the door at the end that nearly did it for me.’
The hilarious footage was first tweeted by BBC producer Julia MacFarlane, who promptly deleted it 20 minutes later.
She wrote: ‘When the kids interrupt you in the middle of live TV…A lovely moment and masterfully handled by our guest this morning on South Korea’.
A BBC spokesman told MailOnline: ‘We’re really grateful to Professor Kelly for his professionalism. This just goes to show that live broadcasting isn’t always child’s play.’
A highly respected expert on South Korean politics, Prof Kelly has written for outlets including Foreign Affairs, The European Journal Of International Relations and The Economist.
He earned his bachelors degree in political science from the University of Miami and completed his PhD at Ohio State.
Kelly moved to Korea in 2008, and married Jung-a Kim, a former yoga teacher who is now a stay-at-home mother to their two children, Marion, four, and James, nine-months.