The Americans fired a 9-under 63 at the Taiheiyo Club's Gotemba Course to close within three shots of the lead at 19-under 197. They had a chance to pull even closer at the par-five 18th, but Duval dumped his second shot in the water fronting the green and Woods lipped out his four-foot birdie attempt.
Host Japan, represented by Shigeki Maruyama and Toshi Izawa, kept the home crowd happy by posting a 65 in the shadow of Mount Fuji. They finished with a share of third place at 18-under with Ernie Els and Retief Goosen of South Africa, Angel Cabrera and Eduardo Romero of Argentina, and Thomas Levet and Raphael Jacquelin of France.
The South African, Argentinian and French teams each turned in 63s on a day rife with low scores. Twenty of the 24 teams in the field shot 6-under-par or better.
The Americans, five shots back at the start of the day, vaulted into the mix with 10 birdies in the third round. Woods led the way with six birdies, including four on the inward nine.
'I think we played fairly well today,' said Woods, the top-ranked player in the world. His British Open-winning partner is ranked No. 3.
'It wasn't a great round but we moved up the board and put ourselves in a position to win,' Woods added. 'I just wish I would have made that birdie putt on the 18th because I don't like to finish that way.'
After Duval made the last of his four birdies at the 11th to take his team to 16-under, Woods took control with two straight birdies. He followed an eight-foot putt at 13 with a sensational 9-iron approach to tap-in range at the 14th.
They both hit irons off the tee at the par-4 15th but their bids for accuracy backfired when Duval's drive finished in the left rough and Woods' shot landed in the right fairway bunker. Duval then left his second short of the green and Woods sent his approach from the sand over the putting surface.
Although Woods threaded the needle between some trees with his punch third shot, his ball rolled 20 feet past the hole and his par try caught the edge of the cup and spun out. Duval had a chance to save the day but missed his putt for par after chipping to six feet.
The U.S. regrouped right away, with Woods birdieing the 16th with a six-footer before knocking his tee shot at the par-3 17th to 10 feet then rolling in the putt to get to 19-under par.
Campbell, No. 30 in the world, took the Kiwis to 20-under with a six-foot birdie at the 13th. The unheralded Smail, a regular on the Australasian Tour with experience playing Japanese courses, got into the act by holing a birdie putt from off the edge of the 14th green.
Campbell missed birdie attempts from eight and 10 feet down the stretch but two-putted from 15 feet at the last for a closing birdie.
'It will be a big thrill tomorrow to play with Duval and Woods,' said Campbell, who finished third behind Woods at the Johnnie Walker Classic in Thailand a year ago Monday and second to him at the TPC of Europe in May.
New Zealand is in position to win its first ever World Cup title, while the U.S. is chasing its third straight Cup and 24th overall since the event's inception in 1953.
Els and Goosen, a duo with three U.S. Open titles between them, is seeking a third victory in this event for South Africa. Els, who captured the 1996 World Cup with Wayne Westner, drew his team within four shots of the lead Saturday with a short eagle putt at the 18th hole.
Romero drained a lengthy eagle putt at No. 18 to move Argentina into third place, as well, while Welshmen Phillip Price and Mark Mouland shot a 62 for a share of 17-under 199 with Sergio Garcia and Miguel Angel Jimenez of Spain (65), Thomas Bjorn and Soren Hansen of Denmark (65), and Andrew Coltart and Dean Robertson of Scotland (66).
Robertson chipped in for birdie at the 15th to lift Scotland to 18-under but he and Coltart ran into trouble at the next hole and were lucky to come away with a bogey. Robertson had an opportunity to make up for the bad hole but missed a four-footer for birdie at the 17th.
The Norwegian tandem of Per Haugsrud and Henrik Bjornstad fashioned an 11-under-par 61 -- the low round of the tournament -- to jump into a tie for 11th with England (63), Australia (64) and Ireland (64) at 16-under 200.
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