Contesting his 400th Grand Slam match, Federer won the first two sets against Norwegian promise Casper Ruud 6-3, 6-1, before being pulled into a tie-break in the third set, but managed to go through with a score of 10-8, Xinhua news agency reports.
"The third set could have gone either way. He had chances early, then later, and then again after," admitted Federer. "It was nice to see him hang in there."
Federer will face Leonardo Mayer from Argentina for a quarter-final spot, and a potential semi-final clash with Nadal looms ahead.
Nadal, in quest for a record-extending 12th French Open title, overcame Belgian 27th-seeded David Goffin 6-1, 6-3, 4-6, 6-3.
He said: "I played against a top player that had injuries, and he's a little bit lower in the ranking now, but when he's playing well, he's a player that already showed to everybody that he can win against every player in this world."
His next opponent will be another Argentinian Juan Ignacio Londero.
Pliskova lost 6-3, 6-3 to Martic, and her exit means that Naomi Osaka is set to maintain her world No. 1 status after the tournament.
She conceded 28 unforced errors, double that of her Croatian opponent, and also lagged behind in points won on first and second serves.
Prior to Pliskova, other challengers to Osaka's throne have fallen by the wayside, as fifth-seeded Angelique Kerber was eliminated in the first round, fourth seed Kiki Bertens was laid low with an illness, and No. 6 seed Petra Kvitova withdrew through injury.
"The court is pretty slow, so you just have to play some extra shots. She has this kind of game which you don't feel good about, and she played well," she added.
Elsewhere, the 2016 winner Garbine Muguruza upset ninth-seeded Elina Svitolina 6-3, 6-3. No. 7 seed Sloane Stephens also marched on with a 6-3, 5-7, 6-4 victory over Polona Hercog.
The Chinese women's pair of Duan Yingying and Zheng Saisai won their doubles second round match over Anna-Lena Groenefeld and Demi Schuurs 6-3, 6-2.
( With inputs from IANS )