The Danes started the final day in Melbourne with a four-shot cushion, and seven birdies, combined with a single bogey, gave them the win on 20 under.
"It's difficult to describe my feelings," said Kjeldsen, before lifting the Hopkins Trophy.
France, China and the USA tied for second on 16 under.
"We both came into this week in good form and we just gelled so well," added Kjeldsen, who holed a 20-yard putt on the last.
"A friendship has been built as well, and I think that's the whole point of the World Cup of Golf."
Victory for Kjeldsen came at the sixth time of asking, while Olesen was playing in his third World Cup as Denmark beat their previous best finish of second in 2001.
French duo Victor Dubuisson and Romain Langasque led the chasing pack with a nine-birdie 63 to share second place with China's Wu Anshun and Li Haotong (65) as well as American pairing Rickie Fowler and Jimmy Walker (66).
Ireland's Shane Lowry and Graeme McDowell finished 11 under, level with defending champions Australia, who were represented by Adam Scott and Marc Leishman.
England's Chris Wood and Andy Sullivan closed with a 65, their best round of the tournament, to finish on nine under.
Russell Knox and Duncan Stewart of Scotland ended on six under while Wales' Bradley Dredge and Stuart Manley hit a final-round 63, moving them to five under.
