Bjorn, who had a 36-hole total of 8-under 134, made a testing six-foot putt to save par on the 16th and a birdie on the 17th before bogeying the final hole after a misjudged approach shot.
American Kevin Streelman was in second place after shooting a 69. He and Matt Kuchar led the overall team standings for the U.S. after 36 holes, with a three-stroke margin over Denmark and seven over Japan and Australia.
Tied for third in the individual stroke-play competition were Jason Day of Australia (70) and Ricardo Santos of Portugal (69). Kuchar was in a four-way tie for fifth after a 68, five strokes behind Bjorn.
Adam Scott, who has won the Australian PGA and Australian Masters over the past two weeks, shot 68 and was nine strokes behind Bjorn in a group with Graeme McDowell (71).
There was no 36-hole cut in the 60-man field that includes 26 two-man national teams and eight individual golfers.
"A few things happened on the golf course today that really tested me," Bjorn said. "I three-putted the first — I thought the greens were a lot firmer than they were early on."
Bjorn shot 66 on Thursday and shared the first-round lead with Streelman.
"I patiently worked my way into the round, maybe not as good as yesterday," he said. "But I played some really good stuff on the back nine, and I've got to take that with me into the weekend."
Streelman, representing the U.S. for the first time, birdied his first four holes but gave three strokes back with a bogey on the eighth and a double-bogey on the ninth, blading a shot out of the bunker.
"I kind of just did what you are supposed to on those first four holes," Streelman said.
"I hit it in the wrong part of the golf course on No. 8 and No. 9, and Royal Melbourne will bite you when you do that which is what makes it such an awesome golf course."
He made a four-foot par save on No. 17, and again on 18.
"That was about a cup out, just breathe on it type of putt and it's one of those when I hit it I knew I had made it," he said of his putt on the 17th.
Streelman, playing Royal Melbourne for the first time, said he couldn't believe the quickness of the greens.
"It is the same feel of Augusta on a weekend — actually I wouldn't know because I haven't made a cut in the Masters yet. So Friday afternoon at Augusta it feels like," he said.
Streelman said despite the change in format here, in which individual stroke-play is being accentuated with US$7 million in prize money, with only US$1 million for the team component, he wants to win the World Cup team trophy for the U.S.
Kuchar and Gary Woodland won the last team event in 2011 in China. The individual portion of the tournament was introduced this year to mirror the format and rankings system which will be used in 2016 when golf makes its return to the Olympics at Rio de Janeiro.
"I am definitely keeping my eye on that team score," Streelman said. "I know he (Kuchar) will be there for me and then hopefully I can continue to play well and we can bring it home."
Day, who birdied No. 15 and then made three good pars to finish the round, didn't have any luck on the greens.
"I had three lip-outs, pretty harsh ones, but definitely happy with shooting 1 under," Day said. "Being out here at Royal Melbourne with how hard and fast the greens are, it can definitely go south pretty quick."