Woods had beaten Rory McIlroy 2&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.
Woods ended the match after McIlroy almost holed a long birdie putt at the 17th.
"We've played tournaments together, battled each other down the stretch at event, but never in a match-play situation", Woods said. Needing to defeat Cantlay to move on, Tiger was down two after eight holes.
Both saw birdie opportunities at nine pass them by following excellent approaches, but it was at the next hole that McIlroy appeared to lose his grip on the contest.
Bjerregaard was able to stay close, and converted birdies on both the 12th and 13th holes, staying just one back.
"I played pretty consistent the entire day", Woods said.
He won his semifinal match when Molinari three-putted from 25 feet above the hole with a helping wind, missing his par attempt from 8 feet.
Elsewhere, Francesco Molinari enjoyed a far more serene round to a semifinal meeting with Kevin Kisner. Woods, buried in a bunker off the tee, hit his third shot before McIlroy had to hit his second.
The three-time Masters victor will spend the next three or four days in Augusta and says he's looking forward to the chance to work on his game.
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Woods had mounted a comeback against Patrick Cantlay, taking a 1-up lead ahead of the 13th hole.
The misadventures not only cost McIlory a chance to tie the match but put Woods 2 up and on the brink of victory.
"We read it inside left center and it went left", Woods said. I think he made two birdies today.
"I'm going to have to go out there and play my own game and not really look at what he does", he explained.
This is the second consecutive year Kisner has advanced to the championship match. Kisner won the round 1 up on his way to the final.
Playing him or beating him? It was just one of those mistakes that Sergio made.
He was only about 100 feet from the hole, but McIlroy pitched long, over the green, and with spectators scurrying out of the way, the ball came to rest against a wooden bulkhead.
A missed putt from four feet on the 18th hole cost Tiger Woods a place in the semi-finals of the WGC matchplay championship at Austin, Texas. McIlroy blasted out and then conceded - with Woods not even having to hit a putt.
In the first match play meeting between the pair, 2015 victor McIlroy missed a series of putts to the right and three-putted the 10th to go three down. He did fall to Brandt Snedeker.