The 23-year-old made seven birdies and an eagle to match the best score on Le Golf National's Albatros course held by Rasmus, Italian Guido Migliozzi and former Argentinian player Eduardo Romero.
Rasmus Hojgaard's 62 came in the opening round of the 2022 French Open, when he was eventually pipped to the title by Migliozzi's own record-equalling effort on the final day.
"I was there in '22 when he was close to winning here," Nicolai Hojgaard said after moving within three shots of co-leaders Xander Schauffele and Jon Rahm heading into the fourth round.
"So I saw it all, how he handled things and how the course was playing.
"It's playing pretty similar. Cool to see that I convinced myself to play a really good round today and see myself hit good shots coming in after a couple of tough days."
Hojgaard was nine shots off the pace after starting the week with a pair of one-under 70s.
He had a 15-foot putt on the 18th green to beat his brother and take the record outright, but saw it slip by the hole.
"I had it in mind when I hit the putt and thought I hit the putt correctly. Just stifled me in the end but obviously pleased with the finish," he said.
"You're going to feel the nerves. It's going to be fun. That's part of it and if you don't like it, it's going to be a tough career and we absolutely love playing when it matters the most."
Thorbjorn Olesen, the other Dane in the field, also enjoyed a strong day, shooting a 66 to climb the leaderboard.
Hojgaard said having Thomas Bjorn, who led Europe to a crushing Ryder Cup victory at Le Golf National as captain in 2018, helping out the Denmark team in Paris has been vital.
"It's huge," added Hojgaard who was part of the winning European team last year on his Ryder Cup debut in Rome.
"He's a guy who has been a big part of my golf career and a big part of Thorbjorn's, as well... He's obviously a huge character to have around. He was very successful in 2018."
Bjorn played down the importance of his involvement, saying: "I took him to a good restaurant yesterday that he liked. That's about (all) the credit I'll take."
Hojgaard, who is bidding to become Denmark's first gold medallist at the Paris Olympics, will play alongside Hideki Matsuyama and Rory McIlroy in the penultimate group in Sunday's final round.
He will tee off tied for fourth with Japan's Matsuyama on 11-under for the tournament.
