The Japanese player led playing partner Fleetwood by one stroke heading down the last, but hit his third shot on the par-four into the water before holing an excellent 12-foot putt to avoid a triple.
Matsuyama, the overnight leader, had to settle for a three-under-par 68 and an 11-under total at Le Golf National.
Britain's Fleetwood missed a par putt on the final green to move atop the leaderboard on his own, but had already done enough to surge into contention with a seven-under 64.
The 33-year-old, who has never won a major title despite a series of near-misses, showed off his expertise around the Albatros course with an eagle and six birdies.
Fleetwood won a record-breaking four points alongside Francesco Molinari in Europe's 2018 Ryder Cup win at the same venue, where he was also victorious in the 2017 French Open.
Reigning champion Schauffele is in perfect position to continue a golden year which has already delivered his first major triumphs, at the PGA Championship and the British Open.
The American added a five-under 66 to his opening 65.
"If you do hit a bad shot, it's easier to accept it because you know you're playing really good golf and you can make it up somewhere else," said Schauffele.
Two-time major champion Jon Rahm also carded a 66 to sit just two shots off the leading trio in fourth place.
"It has been a great day. I didn't need anything spectacular," said the Spaniard.
South Korean rising star Tom Kim, Belgium's Thomas Detry and Taiwan's Pan Cheng-tsung, who won the bronze medal at the Tokyo Olympics, are tied at eight-under as just three shots separate the top seven players.
Kim signed for a 68 after his first-round 66. The highlight of his day was a perfect bunker shot on the third hole which dropped into the cup for eagle.
Scheffler, McIlroy lurking
Scottie Scheffler and Rory McIlroy both struggled to consistently find their best form, but battled hard to stay in the medal hunt.
World number one Scheffler slipped back with a double-bogey on the seventh hole, but played the back nine in four-under and lurks just five shots behind.
McIlroy endured an up-and-down day, making eagle after hitting the flagstick with a sensational second shot on the par-five third hole, but could not avoid mistakes.
An errant drive on the 17th led to a double and left him in danger of falling well adrift, but he hit back with a birdie on the treacherous last to finish the day on five-under.
