The 34-year-old Nadal, seeded third, raced through the first two sets before eventually clinching a 6-2, 6-3, 7-6 (7/3) win over his Australian opponent.
"It was a very difficult match with a very good player like Alexei. For me it's always unique and special to play here," he said.
The Spaniard could usurp Roger Federer at the top of the all-time men's list with a 21st Grand Slam singles title in Paris.
He started in ominous fashion for his rivals and has still not lost a set at Roland Garros since beating Dominic Thiem in four sets in the 2019 final.
Nadal will take on Frenchman Richard Gasquet, against whom he has won 16 times in as many matches, in the second round.
Questions had been asked about Nadal's form earlier in the clay-court season when he lost to Andrey Rublev in the Monte Carlo Masters and Alexander Zverev in Madrid.
But he hit back with the Italian Open title, beating Novak Djokovic in the final.
He carried that form into the first two sets against Popyrin, striking 18 winners past the 21-year-old.
But the youngster looked in prime position to force a fourth set, only to miss two set points on his own serve as Nadal showed off his famous powers of recovery.
The reigning champion powered 6-1 clear in the tie-break before securing a second-round spot at the third time of asking.
"I was lucky to win that third set, I don't know how (I did)," added Nadal.
"The beginning of the match had been very tough too. In these warm and fast conditions it's always very difficult. He's young and the speed of his ball is very strong."
It was Nadal's 101st career win at the French Open, against just two losses -- to Robin Soderling in 2009 and Djokovic in the 2015 quarter-finals.