He remained poised after a double-bogey at the second hole, and roared back with five consecutive birdies to regain control of the $7 million event.
At 10-under 203, he will take a two-stroke lead over Englishman Chris Wood (67) into Sunday's final round.
Spanish big guns Garcia (64) and Jon Rahm (68) are three and four strokes behind respectively.
Kinhult knows victory on Sunday would be career-changing, and hopes his Saturday experience of playing with the lead will hold him in good stead.
"It's going to be nerve-wracking, the first few holes, at least," he said.
"I got off to a shaky start today. Hit some bad swings to be honest, a little bit nervous, feeling the pressure."
Garcia moved into contention with the best round of the week while Rahm sank a 25-foot birdie at the last after hoisting a high eight-iron approach shot over the water hazard guarding the green.
Garcia, the 2017 U.S. Masters champion, plundered the demanding course for eight birdies until his lone bogey at the par-four 18th, where he unexpectedly drove into a water hazard.
"It's not the tee shot I wanted to hit," he said.
"I would love to have not made any bogeys. It's always nice, and on a golf course like this its even nicer."
Garcia is making his first start at Le Golf National. He is using his appearance partly as a reconnaissance mission on a course that will host the Ryder Cup in three months time.
More immediately he will have his sights set on Kinhult, who finished the day with the same lead as he started.
He was in danger of falling back into a tie with Wood after pulling his drive into the brutal hay-like rough at the par-four 17th.
But Kinhult took his medicine and slashed his recovery shot back into play, before getting up-and-down from 140 yards for a superb par, his feat witnessed by only a small gallery, as many fans had departed to watch France's World Cup match on TV.
(Reporting by Andrew Both in Cary, North Carolina, editing by Ed Osmond)
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