With four straight birdies and an unlikely 25-footer to close out the front nine, Woods was 3 under par for his round, two back of the leaders and headed for a few easier holes on the back nine with a chance to put himself squarely in the mix for the final round of the Quicken Loans National.
The fightback fizzled when Woods missed a six-foot birdie putt at the par-5 10th, then missed the green on his approach and made bogey at 11.
But Ancer went even better, matching the course record with an eight-birdie 62.
While he did indeed make some lengthy putts he still missed 13 from inside 10 feet, ranking last of those players who made the cut.
In comments after his round, Woods said he was frustrated he didn't end up 10-under, which he thought he could have achieved on the back nine. "I figured I had to get to at least 10 and making four bogeys definitely hurts that".
Woods meanwhile remained confident about winning his first title since 2013.
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"I know it's boring and everyone says the same but it's the way golf works". "But once again, I did my best".
"I was hitting the ball well", said the 27-year-old. "It's just a matter of keeping focus out here in this heat".
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"I grew up watching him play when I was in my teens and starting to think about turning pro", Molinari, 35, said of Woods.
He struggled early on the West Coast, failing to post a top-10 finish, and as the heart of the European Tour schedule approached he was forced to make a hard choice.
He didn't, and now has more ground to make up.
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He pumped both arms while clutching a new mallet putter when the ensuing birdie bid fell in the heart of the cup to complete Woods' longest streak of consecutive holes below par this season and the most since his first round of the 2015 Quicken Loans National. He won the National in 2009 and 2012 at nearby Congressional Country Club.
"One of the neat things about playing an Open Championship - they don't really care what par is", Woods said. "I just got rolling after that".
"The putts I missed, I hit a lot of good ones, which I don't mind because I hadn't been doing that for the better part of two months". And with a birdie on the final hole for a 67, he tied for sixth to earn his first trip to the British Open.
Six birdies, three of them from over 16 feet, against a bogey, saw Lahiri card 65 and rise to 4-under 206 and give him a chance to register another good finish this week.
Woods has won three Opens in his career, two at St.Andrews in 2000 and 2005 and a very emotional third Claret Jug in 2006 at Royal Liverpool.
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Following a 275-yard drive at the par-4 fourth leaving 150 yards to the flagstick, Woods stuck his approach within seven feet and putted into the center of the cup, eliciting cheers that could be heard several holes away.