The Danes began the final day in Melbourne with a four-shot advantage over second-placed America and the duo were able to hold their nerve.
Kjeldsen and Olesen hit seven birdies and a single bogey to claim the overall victory with a final score of 20-under.
France, China and the USA were all tied for second on 16-under as Denmark beat their previous best finish of second in 2001.
Kjeldsen said: "It's difficult to describe my feelings.
"We both came in this week in good form and we just gelled so well.
"A friendship has been built as well, and I think that's the whole point of the World Cup of Golf."