South African Tim Clark hit a final-hole eagle as he and Fiji's Vijay Singh defeated Lucas Glover and Stewart Cink to claw back vital points for the international team.
Woods took double bogey, and only an eagle on the final hole kept the damage to a minimum.
Oosthuizen came up short in the playoff, but had an unforgettable moment that was replayed over and over with his double-eagle on the second hole on Sunday.
McIlroy was headed toward a missed cut in 2010 until he made eagle on his 16th hole of the second round to make the cut on the number.
Meanwhile Rich Beem overcame a rocky start with an eagle on the ninth hole and a 69, giving him his best position in a final round since February.
The Scot rounded off a memorable day by holing a 40-yard bunker shot for eagle at the final hole to finish on 21-under par and move a step closer to a first European Tour win since 2004.
The first double-eagle ever on that hole, and the fourth in Masters history, instantly vaulted Oosthuizen from third place at seven under par to first at 10 under par the same figure that he and Watson finished at to force the playoff.
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Starting on the back nine, the world No. 32 went to the turn in 31 strokes with three birdies and an eagle three at the 14th hole.
Thompson holed a 50-foot eagle putt on the third hole, relied on a superb short game around the toughest part of the golf course to build a four-shot lead, and hung on for a two-shot win over Geoff Ogilvy that takes him places he always wanted to be.
McGrane, world ranked 250, carded seven birdies and an eagle, with only a final hole bogey spoiling his round.
She needed an eagle at the par-5 final hole but had to settle for par.
Beem compiled seven birdies and one bogey at the Victoria Golf Club, while Melbourne-born Allan's 64 included four birdies and an eagle at the par-five 18th hole.
He knocked the shot about 25 feet from the hole, and almost made an eagle.
WSJ: Masters 2012: Bubba Watson Miracle Shot Lets Him Trump All Comers
The highlight of Mickelson's round was a 21-foot eagle putt at the par-five fifth hole.
The eccentric Frenchman, in joint third overnight, lept to eight under and a share of second place after his eagle feat at the par-five fourth hole.
The Indian signed off his round of eight-under 63 on Saturday by rolling in a 40-foot eagle putt on the ninth, his 18th hole, to move to 15 under par.
After a dramatic 86-foot eagle putt five holes earlier and a birdie on the last hole, he is six under par for the tournament, tied for eighth place in a field of 132 players.
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The first came on the par-five second hole Sunday, when Oosthuizen scored a rare double-eagle.
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Englishman Rose dropped a shot at his final hole, while American Haas had earlier finished strongly with an eagle at 16 -- emulating Woods' opening-day effort.
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He made a 17-foot birdie putt at the 221-yard second hole, and was 7 under his final eight holes with an eagle-birdie-birdie finish.
Driscoll had joined Johnson in the lead with one hole left to play, seven under for the day after five birdies and an eagle at the par-five 14th and he moved into the lead with a 15-foot birdie putt at the last.
He dropped a shot at the opening hole and then double-bogeyed the third, before a birdie at the seventh and an eagle three at the ninth got him back into contention.
Playing in wind gusting to 25 to 30 mph, Webb had two birdies, an eagle and a bogey, capping the round with a 5-foot birdie at the par-5 final hole.
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