Goosen reached the turn at two-under for the day, 11-under overall, to take the tournament lead from India's Randhawa, who birdied his first two holes before falling back to nine-under with a double bogey at the fifth and another dropped shot two holes later.
World number one Woods, who shot a course record 64 on Friday, opened his round with a bogey after a wayward drive off the first tee but redeemed himself at the next hole with a second eagle of the week at the Sheshan International Golf Club.
The American dropped two shots when he found the sand at the par-4 seventh and looked in trouble at the next as well, twice hitting into the rough before a beautiful chip-in from off the green gave him a birdie.
The 12-times major winner shared third place on eight under with Ireland's European Order of Merit champion Padraig Harrington and South Korea's Yong-Eun Yang, who were playing together two groups ahead of Woods, Goosen and Randhawa.
In contrast to the nervous-looking Randhawa, twice U.S. Open champion Goosen played his trademark steady game, picking up birdies at the second, sixth and eighth holes with a single blemish on the front nine.
The $5 million event is sanctioned by the Asian, Australian and Southern Africa tours as well as being the first stop on the 2007 European Tour.