Woods, the top seed in the Bobby Jones bracket and defending champion, will take on Stephen Leaney, the No. 8 seed in the Ben Hogan bracket. Leaney moved on with a 1-up victory over Ian Poulter in the quarterfinals.
The other semifinal is set with Darren Clarke, the three seed in the Gary Player bracket, against Davis Love III, the No. 1 player in the Sam Snead bracket.
The semifinals will be Saturday afternoon with the 36-hole final on Sunday.
Clarke earned his spot in the semifinals with a 5-and-3 beating of Jerry Kelly while Love played in the best match of quarterfinals, dispatching Phil Mickelson, 1-up.
The Love/Mickelson match featured 13 holes won and only five halved. Love won the 14th when Mickelson went left with his approach but Mickelson squared the match at 17 when he knocked an 8-iron from 190 yards to six feet. He cashed in on the birdie putt and it was off to 18 to decide the contest.
At the par-5 closing hole at La Costa Resort & Spa, Mickelson hit a long drive into the fairway while Love drove well right of the short grass. Love laid up and barely made it back to the fairway but Mickelson had a tough decision and pulled his fairway metal from the bag.
Mickelson badly pulled his second right of the green and Love knocked a 6-iron 25 feet from the hole. Mickelson had a tree in front of his pitch but elected to try and hit over it and the ball struck a branch and came down.
Mickelson chipped through the green on to the back collar but Love first putted his birdie try about two feet from the cup. Mickelson needed to hole his putt to extend the match and he did not, falling 1-down.
'I felt like I was in the driver's seat after Davis missed the fairway and had to lay up,' said Mickelson. 'I pull-hooked it and made six. It was a disappointing day. I felt like I was playing better than that.'
Many golf fans hoped for Woods and Mickelson in the final but only Woods did his part in the quarterfinals.
Woods took his first lead with an eight-foot birdie putt at the third and added to his margin with a par at the fourth. Harrington hit a tree off the tee and his drive only went 40 yards.
Harrington won No. 5 but Woods won the seventh with a long birdie putt and the par-5 eighth with a two-putt birdie. The defending champion won the ninth with par after Harrington lipped out a par try on the right side.
Woods, now with a 4-up lead, conceded the 11th to Harrington then dropped the 15th after a poor bunker shot that left him with 30 feet for par. Harrington had a decent look at birdie on 16 but his putt did not find the bottom of the cup.
Woods struck an awful putt from 25 feet at the 17th. The ball died right but the No. 1 player in the world drained the nine-footer to save par, halve the hole and win the match.
'I just hung in there,' said Woods, who is the only player who has won all four World Golf Championships events. 'Paddy made a couple mistakes. I went ahead and made a couple of putts. Overall, it was a good match.'
Leaney, who knocked off one seed Mike Weir in the second round, established a 3-up lead through 13 holes but Poulter mounted a comeback. The young Englishman won the 14th then drained a five-footer for birdie at the 15th to cut the deficit to 1-down.
The match came down to 18 and both players had looks at birdie. Leaney went first and his putt from 25 feet narrowly missed the hole. Poulter had 10 feet and his putt grazed the cup but fell off left, giving Leaney the match.
'I hit it on the line I wanted to hit but it broke left,' said Poulter, referring to his birdie chance to extend the match at 18. 'I didn't think it was going to move at all but here we are.'
Clarke went 2-up through four holes but Kelly, who knocked off Vijay Singh, the No. 1 seed in the Player bracket, in the second round, clawed back. He chipped in for birdie at five then squared the match at six when Clarke lost count of his strokes and conceded the hole.
Clarke won seven with a par then chipped in for eagle at the eighth to go 2- up. He made his second eagle of the round at the par-five 11th when he sank a 12-footer and that gave the former champion a 3-up advantage.
Kelly could not catch a break as Clarke then sank a 20-footer for birdie at the 13th to go 4-up. Kelly landed in a bunker with his approach at 15 and when his blast from the trap did not go in, Kelly conceded the match to the European Ryder Cupper.
'This course has been kind to me in the past and it's certainly been that way again so far this week,' said Clarke. 'I'm just looking forward to playing again this afternoon.'