Oosthuizen, the 2010 British Open champion, came into the weekend with a five-stroke lead over the field, but struggled with his putting and had a 70.
The South African had a chance to retake the lead with a 15-foot birdie putt on the 18th hole, but missed by inches – the last of many near misses throughout the day.
Westwood and Oosthuizen were at 18-under 198. Phil Mickelson (66) made seven birdies and was three shots back.
Ernie Els briefly surged into a share of the lead on the back nine before dumping his ball into a reservoir on the tricky 15th hole and taking a double bogey. He finished with a 69 and was tied for fourth with Bill Haas (66) and Ian Poulter (65) at 202.
Westwood, a former world No. 1, has long been considered one of the most consistent golfers, yet he's never broken through to win a major, finishing runner-up twice – at The Masters and British Open in 2010.
He's also played in more World Golf Championship events than any other golfer (41), but his best result is second place on three occasions, most recently at the HSBC Champions in 2010 when he lost to Francesco Molinari by one stroke.
'It does surprise me that I've played more than anyone else. I didn't realize I was that old,' the 39-year-old Englishman said. 'I know I've played a lot. I've never won one.'
It was the second time in three weeks Westwood shot a 61 - he also had one to beat Charl Schwartzel in the semifinals of the World Golf Final exhibition in Turkey in mid-October.
But as good as his score was Saturday, it wasn't even the low round of the day. That belonged to American Brandt Snedeker, who had a chance at a 59, but just missed a birdie putt on the 18th hole and settled for a 60 – a new course record – and a 203 total.
'You don't get a chance to shoot 59 too often, maybe a couple of times in your career,' he said. 'I'd love to have that putt again.'
Starting the day back in 12th place, Westwood began his charge up the leaderboard with three consecutive birdies on the first three holes.
He then birdied the fifth and seventh holes and sank a 10-foot putt for another birdie on the 573-yard par-5 ninth hole to make the turn at 13 under, four strokes back. The Englishman kept up the pressure on the back nine, closing his round with four more birdies to catch Oosthuizen and Els.
'I just wanted to come out fast and try to get some momentum,' Westwood said. 'That's just been missing from my game the last couple of weeks. I came out quick as you like with three birdies and that got me going and the hole started to look bigger.'
After two rounds of nearly flawless golf, Oosthuizen was five strokes up on the field and looked to be comfortably in control going into the weekend. His two-round score of 16-under 128 was the lowest through 36 holes at any World Golf Championship event since they began in 1999.
But cracks began to appear in his putting game early on Saturday. After making a nice shot from the bunker on the third hole, he missed a routine 5-footer for birdie – the first par-5 hole of the tournament he's failed to make birdie or better.
Oosthuizen then three-putted on the par-3 fifth hole for bogey – just his second bogey of the tournament – and let another birdie chance slip away on the sixth when his 25-foot putt stopped short of the hole by inches. He narrowly missed two more birdie putts within 10 feet on the eighth and ninth holes, settling for par on both.
'Early I didn't play really well and then started to find a little bit of form on the back nine, but just didn't make any putts,' Oosthuizen said. 'The putter went cold on me today.'
While Oosthuizen was missing his putts, Els was sinking his: a 7-foot birdie putt on the fourth, a 3-foot shot for eagle on the par-5 seventh and another 3-footer for birdie on the ninth to pull within one stroke at the turn.
Els' solid round came undone, however, on the 580-yard, par-5 15th hole, which presents golfers with two paths to the hole: a safe route down the length of the curving fairway or a riskier, 250-yard jump over a reservoir straight to the green.
Els took the latter option and paid the price, plunging his second shot into the water. He managed to clear it with his next shot, but then missed an easy 5-footer on the green. He had been bogey-free up to that point in the tournament.
After his remarkable 12-under performance, Snedeker moved up from a tie for 38th place to seventh going into the final day, one shot behind Els. Indian star Gaganjeet Bhullar, ranked 108th in the world, also had a surprisingly strong round, birdieing nine times for a 63. He jumped from 27th place to a tie for eighth.