Price's first-round 67 took him clear of the chasing trio of Mark O'Meara, Justin Leonard, and Bernhard Langer - all on three under par.
Leonard picked up four birdies over the last nine to charge up the leaderboard despite the windy conditions.
The South African Ernie Els made a fearsome start, racing to four-under par after nine holes in front of an appreciative home crowd, but then let a two-shot lead slip with a mistake-riddled back nine.
Scotland's Colin Montgom-erie dropped a shot on the 18th and ended at one under, ahead of the Americans David Duval and Tom Watson, England's Lee Westwood and the pre-tournament favourite Tiger Woods.
Westwood, making his first appearance at the Million Dollar, almost matched Els' start, hitting two birdies in the first three holes, but he then bogeyed the fifth and sixth.
Jesper Parnevik, three under after the first nine, had a disastrous run- in. He dropped a shot on the 15th and another on the 17th, before a double- bogey on the 18th saw him finish the day on two over.
Jim Furyk, finding the going tough in his first Million Dollar event, dropped two shots over the first six holes, and never really recovered, finishing on 75.
Price, last year's winner, was delighted with his performance, despite the wind. "It's amazing. I was telling Tom Watson today that this is a gale for us here, because we normally have a light breeze here," said Price.
"But it certainly helped on the par fives, because we had two of them downwind, No 9 and No 15, so that evened things up a little bit," he added.
The Zimbabwean put his success down to the recent acquisition of a driver.
"It has increased my launch angle by about two or three degrees and it's carrying further.
"At this altitude it hangs in the air longer. It really been an asset to me in the last three weeks," said Price, on a roll after winning the Zimbabwean Open in Harare last Sunday.
Woods, the world No 1, had an indifferent start to his first Million Dollar event, and had to settle for level par going into the clubhouse, having swung between two-over and two-under.
"The wind was going back and forth all day and you just had to make sure you timed it right and stayed committed to your shot," said Woods.
But the 22-year-old is pleased with his form. "It's not too bad," he said. "Obviously it's a little better than it was two months ago. I'm very pleased with the way I'm striking the ball, I'm starting to roll the ball again and it feels like I'm starting to make progress. It feels pretty good."
On playing for a million dollar first prize, Woods said: "I never really think about the money. I always try to get the victory. Get enough victories and the money will come."
Afterwards, Woods praised Gary Player for his role in South Africa's transformation. "When Gary did all the things he did to fight for the end of apartheid, that was significant," said Woods, in the country for the first time.
"I know Gary went through a lot, and was heckled in tournaments he played in and had ice thrown at him. But he spoke out against apartheid. Me coming here after it's all been resolved is not as significant.
"But I've always wanted to play golf in South Africa because of the racial difficulties the country has had.
"It's nice to be here as a person of colour, and it's also nice to see the mixed race galleries as well," added the American.
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