"It would have been the easiest thing for me to just go ahead and pull out of the tournament with what has been going on over the last week," he added. "But I really wanted to come down here and play."
Day's situation wasn't lost on Bjorn.
"Obviously a fraction disappointed, I didn't play that great today," Bjorn said. "But I couldn't be happier for Jason winning. He has gone through a rough time of late and for him to even be here is a big thing and then to go and win a golf tournament ... that's what you want to see."
Day's last tournament victory came at the Byron Nelson Championship on the U.S. PGA Tour in 2010, although he's had four top-five finishes in majors since 2011.
Adam Scott finished third after a 66, three strokes behind. Scott, who was trying to win his third tournament in a row, shot 75 on the opening day, including a 9 on the 12th hole, and spent the rest of the tournament trying to catch up.
Day earned $1.2 million for winning the individual title and helped Australia win the team portion of the World Cup. Day and Scott, who each holed approach shots for eagles Sunday, shared the $600,000 first-place team prize.
American Matt Kuchar shot 71 to finish fourth in individual stroke-play, three behind Day.
Ryo Ishikawa (69) of Japan and Thailand's Kiradech Aphibarnrat, who shot 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 hole for eagle. Bjorn, who was in the rough with his tee shot, made bogey and there was a three-shot swing to put the Australian back in the lead.
Day walked up to the green to pluck the ball out of the hole to the cheers of the roving Fanatics cheerleading squad dressed in Australia's yellow and green, then threw one of them his ball.
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 just 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, but 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 the Masters 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.
Australia finished the team component at 17-under, 10 strokes better than the American team of Kuchar and Kevin Streelman, who finished with a 74 Sunday and was tied for eighth in the individual competition.
Denmark and Japan finished equal third at 5-under in the team event.
