American Matt Kuchar shot a 71 to finish fourth in individual stroke-play, three behind Day.
Ryo Ishikawa of Japan (69) and Kiradech Aphibarnrat of Thailand (70) finished tied for fifth, seven behind the winner.
Day led by four strokes after nine holes thanks to a big swing on the fifth and sixth. Day bogeyed the par-3 fifth after going into bunker and Bjorn birdied, leaving them tied for the lead.
But on the sixth, Day's gap wedge from about 80 yards hit the green once and rolled into the cup for eagle. Bjorn, who was in the rough with his tee shot, made bogey and the three-shot swing put the Australian back in the lead.
On the next hole, Day increased his lead to four over Bjorn when the Danish player three-putted for bogey.
After making the turn with the four-shot lead, thanks to a 12-foot par-saving putt on nine, Day ran into big problems on the 10th when his tee shot went into the left rough. Trying to advance it up the fairway instead of chipping out sideways, he sent the ball but back into the rough.
He chipped back out to the fairway with his third shot, put his fourth on the green and two-putted for double-bogey. That reduced his lead to two shots over Scott and Bjorn, and birdies by Bjorn on 11 and 13 put both players level again until Bjorn's bogey on 16.
Scott, who holed out for eagle with his approach on the first hole Sunday, won the Australian PGA and Australian Masters in his first trip back home since winning at Augusta in April. He'll try to complete the Australian 'Triple Crown' of majors next week at Royal Sydney.
"It's been an incredible day," Scott said. "Thanks Jason, you played so well this week."
The last time the World Cup was captured by a host country was in 1996 when the South African team of Ernie Els and Wayne Westner won at Cape Town.