Woods, the No. 1 seed in the Bobby Jones bracket, and Davis Love III, the top seed in the Sam Snead bracket, will be the only No. 1 seeds to go to the third round as Love bested good friend Fred Couples, 3 and 2.
Vijay Singh, the highest-ranked seed in the Gary Player bracket, got bounced by eighth seed Jerry Kelly, 4 and 2. Mike Weir, the reigning Masters champion and winner of last week's Nissan Open, could not catch up to Stephen Leaney, also ranked eighth, falling by a 3 and 2 margin.
Kelly built a 2-up lead thanks to bogeys by Singh at four and five and held that margin until the 13th. Kelly sank an 18-foot birdie putt at the hole to go 3-up but Singh got one back with a two-putt par at No. 14.
Kelly answered with a short birdie putt at the 15th to go back up by three, then hit it close at the par-3 16th. Singh landed in a bunker and when he failed to hole out, Singh conceded the match to Kelly.
'I really wanted to beat the No. 2 player in the world,' admitted Kelly. 'I'm in great shape right now. I'm looking forward to 36 holes three days in a row, if it turns out.'
The second round was played Friday because heavy rain pounded La Costa Wednesday night and Thursday morning. The plan is to complete the third round Friday afternoon and get the tournament back on schedule, which includes the quarterfinals and semifinals Saturday and a 36-hole final match on Sunday.
Weir and Leaney, teammates for the International side at last year's Presidents Cup, were even through seven holes but Leaney won the next three holes, including a 25-footer for birdie at the 10th, to go 3-up.
Weir won the 11th with a birdie and thought he had a chance at a comeback but Leaney holed out from a bunker for birdie at the 14th to go 3-up. Leaney stiffed his tee ball at 16 and Weir missed his birdie try, then conceded the match to Leaney.
'I just didn't putt very well,' said Weir. 'When I had chances to creep a little closer, the putts I need to make, I just didn't make them today.'
In other second-round matches, Phil Mickelson equaled the record for shortest match at this event, when he trounced British Open champion Ben Curtis, 7 and 6. Mickelson will meet Chris DiMarco, who won the 19th hole to better Stuart Appleby.
Woods moved on to the third round and will face Sweden's Fredrik Jacobson, who advanced thanks to a 5-and-4 thrashing of Denmark's Thomas Bjorn.
'I played well,' said Woods. 'I just need to go out there and do the same thing the rest of the week.'
The other third-round match in the Jones bracket will feature two seed David Toms, who beat PGA Champion Shaun Micheel, 4 and 3, in the second round, against third seed Padraig Harrington, who advanced with a 3-and-2 win over Bob Estes.
Leaney is the highest seed remaining in the Ben Hogan side and will take on 13th seeded Colin Montgomerie in the third round. Monty moved on with a 5-and-4 drubbing of Stewart Cink.
Ian Poulter, seeded 11th, dismantled Duffy Waldorf, 7 and 5, and will battle No. 15 John Huston, who beat Peter Lonard 1-up in Friday's second round.
Mickelson, No. 2 in the Snead bracket, and DiMarco, sixth, are half of the third-round bracket. Love will meet Adam Scott, who beat Robert Allenby on the 23rd hole when Allenby missed a two-footer for par.
Darren Clarke, the third seed in the Player bracket eliminated Alex Cejka, 6 and 5, and will take on Kenny Perry, the No. 2 seed. Perry nearly squandered a lead late but hung on for a 1-up win over Steve Flesch.
The other half of the Player bracket will feature Kelly versus fourth seed Chad Campbell, who disposed of Loren Roberts, 3 and 1.