Oosthuizen, the former Open champion who defeated top seed Henrik Stenson in the second round, thrashed Webb Simpson 5&4 to reach the last eight, where he will face Jason Day after he defeated George Coetzee. Bubba Watson was also knocked out on the 18th by Victor Dubuisson.
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McDowell had reached the third round at Dove Mountain thanks to superb fightbacks against Gary Woodland and Hideki Matsuyama, most notably recovering from three down with three to play to beat Woodland on the 19th.
The former US Open champion was also two down with four to play against Matsuyama before winning on the 18th, meaning the only time he had been in front in both matches was on the final hole.
It was perhaps no surprise therefore that the Northern Irishman lost the first two holes against former Ryder Cup foe Hunter Mahan, the man he beat in the final singles match at Celtic Manor in 2010 to secure overall victory for Europe.
A bogey on the third and par on the fifth was enough to get McDowell back on level terms and he looked like enjoying the novel experience of actually taking the lead on the sixth.
McDowell's tee shot to the par three pulled up just inches away from the hole and was duly conceded, but Mahan produced an excellent approach and holed from 10 feet to be all square.
The Northern Irishman missed several chances around the turn to take the lead and went behind again when Mahan birdied the 11th, but the American then made a mess of the 14th to gift his opponent the hole.
Mahan, who won the title in 2012 and was runner-up to Matt Kuchar last year, swiftly made amends with a birdie on the 15th to take a slender lead. McDowell could not make any ground up on the 16th and 17th, but brilliantly won the last, to take him once again to play-off holes.
Oosthuizen was six under par for the 14 holes completed thanks to an eagle and four birdies. "I played really solid," he said. "The only hole I looked like making bogey was the 10th and I made an 18-20-footer for par.
"I'm putting really well and my ball-striking is improving every day and I am just trying to give myself as many birdie chances as I can."
Oosthuizen was joined in Saturday's quarter-finals by former US Open champion Jim Furyk, who came from three down after six holes to beat rising star Harris English, who had beaten Lee Westwood and Rory McIlroy in the first two rounds.
Furyk, who had never reached the last eight before in 13 previous attempts and has been two down or worse in each of his three matches, will face compatriot Rickie Fowler after he birdied the 18th to edge past Sergio Garcia.
However, the match is likely to be remembered more for a strange incident on the seventh hole when Garcia offered Fowler a half, despite the Spaniard facing five feet for par and Fowler three times that distance.
That was reportedly because Garcia felt bad that he had taken a long time to get a free drop away from a colony of bees on the edge of the green on the previous hole. "I don't want to hit the shot, I've had bad experiences with bees," Garcia was heard to say to the rules official overseeing his drop.
Despite eventually losing by a single hole, Garcia said he had no regrets. "He played much better than me on the last 10 holes and he deserves the win," he said.
"This is a gentleman's game and lately it hasn't felt like it's been like that. This is the way I was brought up by my dad; I felt like my drop on six took too much time. It was the only thing I could do to make myself feel better and not feel guilty."