Against Mayer, who this year reached the final at the Vina del Mar clay-court tournament, Nadal was quick out of the blocks, breaking in the third and fifth games to bag the opening set in half an hour on a sun-drenched Philippe Chatrier court.
He broke to love in the third game of the second set as Mayer seemed to be heading for a humiliating defeat.
Yet the Argentine broke back for 4-4 as Nadal's cross-court passing shot clipped the net cord and went just wide.
Nadal broke decisively in the 11th game with a lob that landed onto the baseline before bagging the set with a stunning forehand passing shot.
The third set was a mere formality and Nadal wrapped it up with a forehand winner.
Earlier, Simona Halep had too many tricks up her sleeve for Spaniard Maria-Teresa Torro-Flor as the fourth seed glided into the fourth round with a 6-3, 6-0 win.
The wily Romanian, the highest seed left in the draw after Serena Williams, Li Na and Agnieszka Radwanska were knocked out early, found perfect angles and defended superbly to counter Torro-Flor's powerful forehand in Paris.
"It's not a surprise because I'm more confident now in myself, and I feel the ball really good here at the French Open," said Halep, who has won more titles in the last 12 months than any other woman except Williams.
Halep, who has dropped only 11 games in three straight-set wins, will next meet American 15th seed Sloane Stephens for a place in the quarterfinals.
Stephens was in less devastating form than in her first two matches but she was still too strong for Ekaterina Makarova in a 6-3, 6-4 win over the Russian left-hander.
The American has shown her consistency on the big stage by reaching the fourth round at six successive grand slam events, the only active woman to do so, but her ability to blow hot and cold on the WTA Tour has prevented her from climbing further up the rankings.
Roland Garros has not been a happy hunting ground for Ana Ivanovic since she won the title in 2008 and the popular Serbian failed to break that jinx on Saturday as she lost 6-3 6-3 to Czech Lucie Safarova.
The 23rd-seeded Safarova, who entered the match with a 4-2 record against the 11th seed having won their last four encounters, was never really bothered by Ivanovic's uncontrolled power.
A dejected Ivanovic said: "It's definitely tough. She's a lefty, it's not something we face every day and the ball comes a little bit differently.
"In the beginning I felt like I had a lot of chances I didn't use. Had I used them, the match maybe would have had a different tone to it."
Safarova will face another champion in 2009 winner Svetlana Kuznetsova after the Russian 27th seed beat Czech fifth seed Petra Kvitova 6-7 (3), 6-1, 9-7 in 3 hours, 13 minutes on court Philippe Chatrier.
Kvitova took two medical timeouts and came back from the first with a heavily strapped right thigh, but it did not seem to hamper her as she peppered the court with winners.
But her 65 unforced errors eventually cost her as she bowed out on the third match point.
Italian 10th seed Sara Errani, runner-up in 2012, won 20 points in succession during her 6-0 6-1 demolition of Israel's Julia Glushko. She set up a meeting with Serbian sixth seed Jelena Jankovic who demolished Romanian Sorana Cirstea 6-1 6-2.