Oosthuizen eagled the par-five second, then added four birdies to tighten his grip on the third-round encounter at Dove Mountain.
"I'm playing some of my best (golf) now," Oosthuizen told reporters. "I'm hitting the ball really good and just need to make the putts.
"I made quite a few birdies coming in, and he put himself in a few tough spots off the tee to make bogeys and pars. My golf has been really solid."
PGA Tour veteran Jim Furyk and fellow American Rickie Fowler also reached the last eight, both after staging impressive fightbacks.
Furyk came from three down after six holes to win a hotly contested match with compatriot Harris English one up, making sure of the win with birdies at 16 and 17 before booking his place in the quarter-finals with a matching par at the 18th.
Fowler, who upset 2010 champion Ian Poulter and in-form American Jimmy Walker in his first two matches, rallied from three down after eight holes to pip Spaniard Sergio Garcia one up.
Fowler piled up five birdies in the last 10 holes to seize control and will take on Furyk in Saturday's quarter-finals.
In other third-round matches, Australian Jason Day was two up on South African George Coetzee after 15 holes and Jordan Spieth was one up on fellow American and defending champion Matt Kuchar after 13.
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Gene Cherry)
(This story corrects headline with Fowler, not Garcia, reaching last eight. No change to story)
