Denmark's dynamic duo will start the final round at Kingston Heath in Melbourne with a four-shot lead after Olesen and the more experienced Kjeldsen continued their rise with a two-under-par 70 in foursomes on Saturday.
To seal the most significant victory in Danish golfing history, the pair now at 14-under must outplay much higher-ranked opponents in the last group — Rickie Fowler (12) and Jimmy Walker (20) of the United States.
The format switches back to the more scoring-friendly four ball play on Sunday.
Fowler and Walker kept in touch with the underdogs by negotiating an equal-best three-under-par 69 to go 10-under overall, thanks to clutch putting from Walker.
Leading by three at the start of play, Denmark appeared to be coasting as they opened a six-shot lead after four birdies and just one bogey through the first 10 holes.
But it looked like the tide could turn when Olesen lost his drive on the par four 11th.
Denmark were forced to drop a ball in deep rough near an out of bounds fence, despite Olesen disputing the official's ruling regarding placement.
It took a brilliant recovery shot and then a pressure putt from Kjeldsen to limit the damage to just a bogey and maintain his team's advantage.
The closest Denmark has come to winning the World Cup was back in 2001 when Thomas Bjorn and Soren Hansen finished equal second to South Africa.
China's pursuit of their first World Cup victory has been the other major story of the week, but it seemed a distant prospect after a double bogey and a bogey on the first two holes on Saturday.
However Ashun Wu and Li Haotong stayed positive to endure a rollercoaster even-par round that kept them at nine-under overall, probably close enough to pull off what would be an incredible triumph.
Hideki Matsuyama and Ryo Ishikawa posted a 71 for Japan, while Victor Dubuisson and Romain Langasque carded a 72 for France to sit equal fourth at seven-under with Spanish duo Rafa Cabrera Bello and Jon Rahm, who scored a 73.
England, meanwhile, will be kicking themselves after a nightmare back nine culminated in a five-over par 77.
That score included four bogeys and a double bogey, adding up to a birthday Chris Wood would rather forget as he and Andy Sullivan dropped from equal fourth overnight to tied 17th.
Saturday's third round all but finished the title defence of the home nation Australia, represented by Adam Scott and Marc Leishman. They will start the final round 10 shots off the pace.