Woods had beaten Rory McIlroy 2 and 1 in their first match play meeting in the last 16 but Bjerregaard completed a 1-up victory over the former world number one when the American missed a short putt on the 18th hole.
"Just to get the chance to play against him today is pretty special," Bjerregaard told PGA Tour Radio.
"I'm pretty proud to say I beat Tiger Woods. Not many people can say that."
The 50th seed will play American Matt Kuchar, who defeated Sergio Garcia 2 up, in Sunday's semi-final.
Bjerregaard, 27, eagled the 16th from 29 feet to draw level with Woods and after both made birdies on the 17th, the Dane won the match at the last to complete a stunning day after he beat Swede Henrik Stenson 3&2 in the morning.
Bjerregaard missed a 17-foot birdie putt at the last only to have Woods lip out for par from four feet to decide the match.
Woods admitted he misread the putt.
"We read it inside left, left centre, and the putt went left," he said. "I went back and hit it again and it did the same thing. Just one of those weird spots."
Bjerregaard's birdie at the first gave him the early lead but three consecutive birdies from the third put Woods 2 up.
"Tiger had a great stretch on the front nine where he stiffed a few and made a couple of good putts," the Dane said.
"So it looked a bit uphill there but I made some great putts coming in."
Italy's Francesco Molinari will meet American Kevin Kisner in the other semi-final at Austin Country Club.
Molinari, the highest remaining seed at seventh, beat American Kevin Na 6 and 5 and Kisner saw off South African Louis Oosthuizen 2 and 1. Na had knocked out second seed Justin Rose in the morning.