The former world number one is the only golfer with multiple individual Olympic medals after winning silver in Rio in 2016 and taking bronze in Tokyo three years ago.
Ko started the day three strokes behind overnight leader Metraux but caught her playing partner by making six birdies to card a 68 and reach nine-under for the tournament.
American rising star Rose Zhang and Japan's Miyu Yamashita are tied for third place, just two shots behind the leading duo.
Reigning champion Nelly Korda briefly threatened near the top of the leaderboard but will start Saturday's fourth round on four under par.
New Zealander Ko led by two shots on the 18th tee, but Metraux leapt back alongside her with an eagle on the par-five closing hole.
Metraux, the world number 137, holed a 20-foot putt to the delight of the thousands of fans around the green, with even Ko raising an arm in celebration as her title rival's ball found the bottom of the cup.
Zhang, a former world number one amateur, charged into contention with a brilliant 67.
She briefly tied the lead with an eagle on the par-five 14th, but hit her approach shot into the water on the following hole and made a double bogey.
But the 21-year-old finished in style, almost holing out for albatross and tapping in for another eagle.
Yamashita also jumped up the leaderboard with a bogey-free round of 68.
Thai star Atthaya Thitikul is one stroke further behind on six-under after a 69, one shot ahead of Colombian Mariajo Uribe who also topped the leaderboard midway through the back nine before dropping back.
"To be in contention for the medal for my country, is like an intense experience to be honest," said Atthaya. "If I can get it, it's going to be really special."
Korda stays in hunt
World number one Korda kept her title defence alive despite another topsy-turvy round that ended in a second successive 70.
The American sits tied at four-under with the Chinese pair of Lin Xiyu, who birdied the last three holes, and Yin Ruoning, as well as home favourite Celine Boutier.
Korda slipped up with two early bogeys before bouncing back with five birdies, sparked by holing a wedge from 82 yards from the rough on the seventh, to close in on the leaders.
But the 26-year-old, who made a quadruple bogey on the 16th hole on Thursday, missed a par putt from two and a half feet on the penultimate green.
Korda also failed to birdie the 18th and will start the final day five shots off the pace, hoping to eclipse compatriot Scottie Scheffler who came back from four strokes behind after 54 holes to win the men's event last week.
"I'm happy with the way that the day played out after the start that I had," she said.
"Obviously I wish I could take 17 back but I can't. I have six three-putts this week, and that's just the reality. I'm not going to be sad about it."
France's Boutier, the first-round leader, dug deep after a triple bogey on the 13th by finishing with three straight birdies.
Australian Hannah Green, who was 10 shots off the lead after the opening day, carded a 66 to move into medal contention on three-under, level with this year's PGA Championship winner Amy Yang.
