The world number one was locked in a unique sudden-death playoff for the trophy with South Africa's world number three Ernie Els, after both teams had ended regulation play level on 17 points.
Woods had to hole an uphill 15-footer to stay in the match.
"It was tough, not only for the fact that if you missed your putt, you let your team mates down, your captain, the assistant captain, all of the wives, girlfriends and everybody that's part of the team," Woods said.
"You let everyone down with one putt, that's a lot of pressure. That was actually one of the most nerve-wracking moments of my life."
The 27-year-old added: "When I was getting over the putt, I kept reminding myself: just a little slow. But I saw all this red (his U.S. team mates) and I was just trying to block that out."
"I just got into my little world and made the putt."
Non-playing U.S. team captain Jack Nicklaus said the putt, which Woods made at the third extra hole, was amazingly difficult.
"We couldn't even see the hole, let alone try to putt," said Nicklaus, who won 18 majors.
"It was up and over, down and sliding away, with a break of 10 to 12 inches."
"It couldn't have been a tougher putt, because not only do you have to play the right break, you have to get the right speed and just the right strength, otherwise it's not going to stay in the cup.
"He just played it perfectly."
Earlier in the day, Woods comfortably saw off the challenge of Els in their crucial singles match.
Els, the local favourite, lost 4 and 3 against an inspired Woods, who played his best golf of the week in sweltering heat at the Links Course at Fancourt.
Woods took the lead for the first time over Els with a two-putt for birdie on the par-five fifth hole and picked up further shots on the sixth, eighth and 12th holes while Els won the 10th.
Woods closed out the match on the par-four 15th when 34-year-old Els, a three-time major winner, three-putted from 60 feet to give the United States a vital point.
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