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.