Day, grieving the loss of eight relatives killed in the Philippines by Typhoon Haiyan, rolled in six birdies to hold a one-stroke lead over Danish overnight leader Thomas Bjorn on a sunny, breezy day at Royal Melbourne.
Day's composure on the sandbelt course's slick greens not only put him into contention for individual honours, but also put Australia a stroke ahead of the United States in the team component.
"Very excited," the world number 18, who sits on a nine-under total of 204, told reporters. "I've just go out there and stay patient.
Sixty players are competing for individual honours for the first time at the biennial World Cup, which was previously solely a team tournament.
Twenty-six two-man teams also competing, with the best aggregate scores after four rounds of strokeplay determining the winning nation.
After two days of patient, solid golf, Danish veteran Bjorn found the going tougher but managed to graft to an even-par 71 to lie outright second, with world number seven Kuchar a further two strokes adrift.
Kuchar, who won the last World Cup in 2011 with compatriot Gary Woodland when it was solely a team event, shot a bogey-free 68 to move into outright third.
"I know my game has been in the kind of shape that gives me a chance to win a lot," said Kuchar, who was runnerup to Scott at the Australian Masters on Sunday.
"Certainly it would mean a lot to be the World Cup individual champion. It would mean a lot to be the team champion."
The rangy 35-year-old's unflinching control made up for a difficult day for compatriot Kevin Streelman, who drifted down the leaderboard to four-under after carding a three-over 74.
The famed sandbelt course has frustrated much of the field throughout the tournament, but Welshman Stuart Manley, one of eight players competing in the individual tournament alone, endured the ultimate emotional roller-coaster on Saturday.
After opening his round with successive birdies, the 34-year-old aced the 161-metre par-three third to soar into second place and mistakenly believed he had won a Mercedes car offered as a prize for holes-in-one scored on Sunday.
After patting the display car and exchanging 'high-fives' with anyone within reach, Manley promptly imploded on the following par-four with a septuple-bogey 11 to crash from a total of seven-under back to even-par.
Manley bounced back, however, to finish with a creditable 72 for the day, seven strokes from the leaders.
"Probably the highest of highs and the lowest of lows," he said. "The Aussie fans are pretty brutal."
Former major winner Graham McDowell, playing for Ireland, shot a 68 to remain an outside chance to take individual honours, six strokes behind.
World number two Adam Scott, left reeling by a horror quintuple-bogey in his opening round, shot a second consecutive 68 to be a further stroke adrift in joint eighth place.
While individual honours may be beyond the U.S. Masters champion, Scott will push for a win for Australia following his pre-tournament prediction that he and Day would win the team competition.
- Sports & Recreation
- Thomas Bjorn