World number three Nadal saved two set points in the third set on his way to a well-earned 6-3 6-2 7-6 (7-3) victory in Tuesday's contest at Roland Garros.
It means that only three players have taken a set from the Spaniard at the grand slam event since 2016 – Diego Schwartzman, David Goffin and Dominic Thiem.
Nadal's victory over Popyrin was his 31st in a row in Paris since 2015 and keeps him on course for a record-extending 14th triumph in the French capital.
The third seed was impressed by what he saw from his 21-year-old opponent, who showed admirable resistance by saving four break points to force a tie-break in the third set.
"With this serve, with these kinds of shots from the baseline, he has everything to become a top player," Nadal said. "If he wants to do it, of course he is going to have his chances, because he has a lot of very difficult things in his game.
"He has everything to become a great, or at least a fantastic player. Let's see. Let's see what is going on in the next couple of months and years."
Asked about his impressive record of rarely dropping a set, Nadal said: "We are playing best-of-five. The other player needs to win two more sets to beat you.
"I know I'm going to be there fighting for every single point. Of course I don't want to lose the set at all, but that's part of the game.
"I'm happy to win that third set. I don't know how I won it, but I did, and I'm glad."
RUBLEV BEATEN BY BIG-SERVING STRUFF
Nadal's path to another French Open title is a little clearer following seventh seed Andrey Rublev's surprise first-round exit to world number 42 Jan-Lennard Struff.
Rublev had reached the quarter-final stage at his previous three majors and was much fancied to go far in Paris, but he fell 6-3 7-6 (8-6) 4-6 3-6 6-4 to the big-serving Struff.
The German hit 25 aces on his way to beating a top-10 opponent at a grand slam for the first time, setting up a second-round meeting with Facundo Bagnis.
"I was feeling fine," Rublev said after the defeat. "I practiced well. I don't know why I lost. I think we played a good level today. I played a solid two sets, third and fourth, and then the fifth set.
"I think I should have won the second set, but this is tennis. I was serving for the set and I didn't make it, then in the tie-break I was a bit unlucky and he played well."
MONFILS ENDS SLAM DROUGHT
Fan favourite Gael Monfils battled back from a set down to beat Albert Ramos-Vinolas 1-6 7-6 (8-6) 6-4 6-4 and earn his first grand slam win since the 2020 Australian Open.
Monfils has struggled since tennis resumed last year after the coronavirus-enforced break, with Tuesday's win on home soil just his second this season.
Elsewhere on Tuesday, Australian Open semi-finalist Aslan Karatsev kicked off his French Open campaign with victory over Jenson Brooksby in three sets.
Tenth seed Schwartzman, who reached the last four in Paris last year, saw off Lu Yen-hsun 6-2 6-2 6-3 in his opening match.
Matteo Berrettini was made to work a little harder for his victory against Taro Daniel, the ninth seed advancing through 6-0 6-4 4-6 6-4.
There was an upset later in the day, though, as number 20 seed Felix Auger Aliassime was beaten in four sets by world number 98 Andreas Seppi.