Woods' next match is set for Saturday against Rory McIlroy, who has a chance to return to No. 1 in the world if he wins the tournament.
During the WGC-Dell Technology Match Play in Austin, Texas, on Thursday, Woods sent the ball over the greens and into the bushes near a tent on the 10th hole.
He then hit an eagle on the 16th hole to make it even, and hit a birdie putt on top of Woods at the 17th to take the match to the 18th.
Equally surprising was how Woods won earlier Saturday against McIlroy, a big match between the two biggest stars left at Austin Country Club.
The three-time WGC-Match Play victor won the par-5 12th before holing out his second shot from 82 yards at the par-4 13th. But Woods made McIlroy putt out from two feet out. and McIlroy missed.
Bjerregaard, who said he had dreamed of playing against Woods since he was 10, relished the challenge of facing him in person and never looked intimidated as he completed a 1-up victory over the 14-times grand slam victor.
"I didn't think it was ever going to come true", he said.
"I just hope I put up a better fight than I did in Atlanta previous year", McIlroy said.
The Italian will play Kevin Kisner, who won three holes in a row to beat Louis Oosthuizen 2&1.
He got onto his knees and played an incredible left-handed shot that not only cleared the ball from under the bush, but also rolled it all the way to within just a few feet of the hole.
McIlroy gave a nod to his final-round duel with Woods at last year's US PGA Tour Championship, where Woods captured his 80th career US PGA victory with a final-round surge while McIlroy struggled.
Having fought back valiantly from three down with seven holes to go to just one in arrears, he produced a stunning 395-yard drive down the par five 16th and had just a wedge in his hands for his approach.
Fresh off an awesome left-handed shot on a ball that was under a bush during Thursday's action, Tiger continued the theatrics on Friday afternoon by sinking an 82-yard wedge shot on the par-4 13th for an eagle 2.
McIlroy hit a shot so bad that it was short and right.
It all happened too soon for Kuchar to have formally conceded which led Garcia to lose the hole and fall two behind in their quarter-final.
Woods followed with birdies at 14 and 16. "And just to get to play him was an experience for me".
"So it looked a bit uphill there but I made some great putts coming in".
McDowell had six birdies, two bogeys and a par on the front nine in mostly calm conditions with occasional showers, then birdied four of the first six holes on the back nine.