Westwood celebrated succeeding Colin Montgomerie as the European Tour's golfer of the year with the day's best round of the elite 12 players competing in the South African event. Montgomerie could only manage a 74 to lie 10 strokes adrift of Zimbabwe's Price, with only the American Jim Furyk behind him.
Price, chasing his third million dollar (pounds 610,000) prize at the Sun City venue, added a 68 to his opening 67 for a 135 total, which leaves him one clear of the Open and US Masters champion Mark O'Meara (67).
Westwood and Justin Leonard (68) are a stroke further back on 137, with Bernhard Langer (70) and Ernie Els (69) sharing fifth spot on 139. Tiger Woods (68) is on 140, with Montgomerie back on 145.
At the Australian Open in Adelaide, Greg Norman slumped to a second-round 82 to survive the weekend with nothing to spare and Justin Rose missed his ninth successive cut since turning professional. Nick Faldo reversed his fortunes with the day's best round, moving to within five strokes of the leader, Paul Gow, with an impressive three-under-par 69.
Norman and Fred Couples finished with two-round totals of 152 - right on the eight-over cut - while Rose shot an 81 - including a double-bogey six on the 18th - to finish at 153, one away from qualifying.
"I didn't have control of the elements today," Norman said. "I lost it and didn't get it back."
Ronan Rafferty will host the first world ice golf championships, to be staged 500 km (300 miles) north of the Arctic Circle in Greenland next year.
The tournament, played with brightly coloured balls and greens sprayed with red dye, is to be held on 27 and 28 March on the sea ice off Uummannaq in northern Greenland.
Scores, Digest, page 25