The big hitters' duel at Sun City was all square over the first four extra holes, but Price's better approach shot on the third visit to the par-four 17th left him with an easier putt.
Woods' putt from the edge of the green swung agonisingly close to the cup. But Price, the 41-year-old defending champion, had done the work with the approach and coolly knocked in a 10-foot putt for his third Million Dollar prize.
The 22-year-old American, playing in South Africa for the first time, had set up the play-off with a stunning 18th-hole birdie to go level with Price at 15 under. "You couldn't have scripted it any better than that. You know, both Tiger and I played very well today, and I knew going down 16 after we'd hit our tee shots there that one of us was going to win it because the guys behind us [overnight leader Lee Westwood and Justin Leonard] weren't really doing anything," Price said.
Westwood, who hit 66 on Saturday to lead going into the final round by two shots, bogeyed two of the first four holes and never really recovered as the field closed in around him.
He finished three shots back, joint fourth on 12 under with Mark O'Meara.
Leonard led briefly, but his challenge faded with a bogey on the 16th, when Price and Woods were on an irresistible surge.
"It was really exciting for both of us. It really was. What a day. I don't know how we're going to top this one," said Price, who won in Sun City last year and in 1993.
He was full of praise for Woods, 19 years his junior: "Like I said to him today, `I've got a great future behind me. Yours is all ahead of you'."
Later Woods, disappointed at losing but encouraged by his surge, felt he had blown it.
"I had a chance. On 17, 18 in the first batch of play-off holes I had a chance. I pulled both putts, right to left, and that's as good as it gets. In order to beat a calibre of putter like Nick Price you're going to have to make one of those putts, and I didn't make one of those putts," he said.
Woods put his uninspired performance over the first two rounds down to jet-lag and he now faces another long flight to Australia for the President's Cup.
Meanwhile, Greg Chalmers won the Australian Open championship after an emotionally charged finish to the final round. Chalmers ended with a closing round two-under-par 70 for a 72-hole total of even-par 288 then watched on as his two nearest rivals both missed birdie attempts which would have forced a play-off.
Stuart Appleby, playing his first tournament at home since his wife was killed in a traffic accident outside a London station, narrowly missed an 18-foot putt on the final hole to finish one shot behind after a final round 72. The 1989 Australian Open winner Peter Senior also missed a long- range putt on the final green in a round of 70 to finish tied with Appleby at one over.
Nick Faldo's expected challenge failed to materialise after he dropped two shots on the front nine to fall five off the pace. The Englishman played the back nine in one-under but was never close enough to mount a serious challenge.
Chalmers, who qualified for next year's US PGA circuit after coming through the tour school last month, became the first left-hander since Claud Felstead in 1909 to win the championship.
He looked to be cruising to a comfortable victory midway through the final day as he thrived in the gusty conditions at the Royal Adelaide links. Then came the dramatic final few holes.
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