Day made up five strokes on second-round leader Thomas Bjorn and has a one-stroke lead going into Sunday's final round. Day is at 9-under 204 while Bjorn is 8-under after shooting even-par 71 Saturday.
Kevin Streelman, who shared the first-round lead with Bjorn and was in second after Friday's play, battled through a 3-over 74 Saturday and slipped into fifth place at 4-under.
Day began his third round with back-to-back birdies. He had a bogey at the par-3 fifth but made the turn at 2-under after a ninth-hole birdie.
He had a birdie at 13 that got him within one stroke of the lead, tied for first with a birdie at 15 and pulled clear with another at 17.
Sixty players are invited to the World Cup of Golf. They take part in both individual and team competitions. The top two players from each country have their scores combined for the team score.
The Australian team of Day and Adam Scott shot into the lead at 11-under 415, one shot ahead of Americans Kuchar and Streelman. No other team is better than 4-under.