Woods was unable to follow up his last-16 win over Rory McIlroy earlier in the day with another victory at Austin Country Club, losing two of his last three holes to be beaten 1up by the 27-year-old Dane.
Bjerregaard will now meet Matt Kuchar on Sunday morning after the American survived a late fightback from Sergio Garcia to win 2up, while Francesco Molinari will face Kevin Kisner in the other semi-final.
“We had a great match and it was a shame it had to end like this,” Bjerregaard said. “Obviously I’m happy to come out on the winning side and I really enjoyed playing with him [Tiger Woods] today.
Bjerregaard made the perfect start with a five-foot birdie at the first and missed an attempt from double the distance at the second to extend to his advantage, only for Woods to birdie his next three holes and race two clear.
Woods failed to convert a chance for a fourth straight birdie from eight feet at the sixth, before a three-putt bogey at the par-three next halved the 43-year-old’s lead.
The Dane briefly levelled with a close-range putt at the 10th but found water off the next tee to allow Woods to win the hole with a bogey, with the pair then exchanging birdies at the par-five 12th.
Bjerregaard pulled level again by draining a 30-foot eagle at the par-five 16th and holed a 15-footer to match Woods’ birdie at the par-three next to leave the contest all-square going up the last.
As Bjerregaard pitched to 15 feet and two-putted for a closing four, Woods found the bunker with his approach and then missed from five feet to save par and hand the 50th seed victory.
Kuchar held a three-hole lead over Garcia until the Spaniard posted three birdies in a four-hole stretch to cut the advantage to one with two to play, only to complete a 2up victory when Garcia conceded the final hole.
Molinari won six of his first seven holes on his way to thrashing Kevin Na 6&5 and set up a meeting with last year’s runner-up Kisner, who defeated Louis Oosthuizen 2&1 – in the other semi-final.
Watch the WGC-Dell Technologies Match Play throughout the weekend on Sky Sports. Live coverage continues on Sunday from 3pm on Sky Sports Golf and 9pm on Sky Sports Main Event.