The trio sit on five-under after 54 holes at Torrey Pines, but both defending champion Bryson DeChambeau and 2011 winner Rory McIlroy are lurking ominously on three-under, with Jon Rahm and Dustin Johnson both within four shots of the lead.
Oosthuizen has been a runner-up in all four majors: the 2015 U.S. Open, 2012 Masters, 2015 Open (in a play-off) and the 2017 and 2021 US PGA Championships.
The 38-year-old got off to an indifferent start on Saturday, making the turn at one-over 36. Another bogey at the 10th pushed him three strokes out of the lead and seemingly out of the conversation.
Then he rolled in a 30-foot birdie putt on the par-3 16th and the monster eagle putt at 18 as he signed for a 70.
“Look, it will help a bit, but I need to play well,” said Oosthuizen, responding to his wealth of major-championship experience.
“There's a lot of great players who have a chance of winning this, and I just need to go out and play as good as I can tomorrow.”
Hughes, who missed the cut in his three previous US Open starts, converted a 63-foot eagle putt on the par-5 13th and closed with an up-and-down birdie from a greenside bunker on the 18th for a 3-under-par 68.
“I don't think I'm ever surprised when I play well,” said the 30-year-old, who is bidding to become the first Canadian to win the event.
“I wouldn't say I necessarily expected to be in the last group this week, but I know that my game is good enough to win on the PGA Tour. I've done it before.
“This is a bigger stage, but again… it's an 18-hole golf course and 72 holes, so I just need to do a lot of the same things I did to win that tournament.”
Henley holed a 65-foot bunker shot on the 11th, statistically the hardest hole on the course, for a birdie en route to a 71 - six shots better than his fellow halfway leader, Englishman Richard Bland.
“Tied for the lead going into the last day of a major, you never know,” said Henley, whose last win was the 2017 Honda Classic.
“I'm 32, I don't know how many more good years I have left, but hopefully a lot. I'm just excited. You always wonder what it would feel like to be in contention.”
McIlroy, who is bidding to land his first major title since the 2014 Open, carded five birdies, including a chip in at the 12th, in his 67, while DeChambeau signed for a bogey-free 68.
Three shots back are Rahm, whose first PGA Tour win in 2017 came at Torrey Pines, Matthew Wolff, the runner-up last September at Winged Foot, and Scottie Scheffler.
Johnson, the 2016 winner, sits four behind along with 2020 US PGA champion Collin Morikawa and local favorite Xander Schauffele.