The 27-year-old follows in the footsteps of Wayne Ferreira, who made the last 16 in 1996, after knocking out Canadian 14th seed Milos Raonic, 7-5, 7-6 (7/4), 6-3 in the third round.
The 23rd-seeded Anderson has equalled his best Grand Slam performance -- a run to the fourth round at the Australian Open this year.
He will face Spanish fourth seed David Ferrer for a place in the quarter-finals.
"Wayne was a great player," said Anderson, who made his first career claycourt final in Casablanca this year.
"I have a long way to go until I match his achievements, but being in the fourth round is great for me."
Ferreira's run in 1996 ended in the fourth round when he lost to two-time champion Jim Courier.
But he remains South Africa's most consistent singles player of recent years, reaching the semi-finals of the Australian Open in 2003.
Anderson has a 1-1 record against Ferrer, a semi-finalist at Roland Garros in 2012.
The Johannesburg native beat the Spaniard in three sets at the Indian Wells Masters this year.
Ferrer had beaten him in straight sets in the US Open first round in 2012
"I will have to play really good tennis, he's one of the best players in the world and on clay he's even tougher," added Anderson.
"But I can look back at when I beat him at Indian Wells and see what I did well then."
Anderson said he was delighted to have taken the sting out of Raonic's big serve, aided by the slow, cool conditions on Court Seven.
"Milos is a very tough opponent. He has such a good serve but I did a great job on his second serve.
"Being in the fourth round is great. People say that clay is not my surface but I just needed more experience."
Raonic was hoping to become the first Canadan man ever to reach the fourth round, but he admitted he had been outplayed by the big South African.
"I lacked execution. I didn't play great but he made me feel uncomfortable," said the 22-year-old.
"He just didn't give me any breathing space.
"He's doing things well on clay. He beats the guys he should do and is only losing to the top guys."
French Open results on Friday (x denotes seeded player):
Men:
Third round:
Tommy Robredo (ESP x32) bt Gael Monfils (FRA) 2-6, 6-7 (5/7), 6-2, 7-6 (7/3), 6-2
Nicolas Almagro (ESP 11) bt Andreas Seppi (ITA x20) 7-6 (7/1), 6-0, 6-4.
Kevin Anderson (RSA x23) bt Milos Raonic (CAN x14) 7-5, 7-6 (7/4), 6-3
David Ferrer (ESP x4) bt Feliciano Lopez (ESP) 6-1, 7-5, 6-4
Jo-Wilfried Tsonga (FRA x6) bt Jeremy Chardy (FRA x25) 6-1, 6-2, 7-5
Viktor Troicki (SRB) bt Marin Cilic (CRO x10) 7-6 (14/12), 6-4, 7-5
Gilles Simon (FRA x15) bt Sam Querrey (USA x18) 2-6, 6-3, 2-6, 7-6 (7/2), 6-2
Roger Federer (SUI x2) bt Julien Benneteau (FRA x30) 6-3, 6-4, 7-5
Second round:
Rafael Nadal (ESP x3) bt Martin Klizan (SVK) 4-6, 6-3, 6-3, 6-3
Tommy Haas (GER x12) bt Jack Sock (USA) 7-6 (7/3), 6-2, 7-5
Mikhail Youzhny (RUS x29) bt Federico Del Bonis (ARG) 6-3, 6-7 (5/7), 7-5, 6-4
Jerzy Janowicz (POL x21) bt Robin Haase (NED) 6-4, 4-6, 6-4, 6-3
Nikolay Davydenko (RUS) bt Denis Istomin (UZB) 6-4, 7-5, 6-2
Richard Gasquet (FRA x7) bt Michal Przysiezny (POL) 6-3, 6-3, 6-0
John Isner (USA x19) bt Ryan Harrison (USA) 5-7, 6-7 (7/9), 6-3, 6-1, 8-6
Stanislas Wawrinka (SUI x9) bt Horacio Zeballos (ARG) 6-2, 7-6 (7/2), 6-4
Janko Tipsarevic (SRB x8) bt Fernando Verdasco (ESP) 7-6 (7/3), 6-1, 3-6, 5-7, 8-6
Women:
Third round:
Serena Williams (USA x1) bt Sorana Cirstea (ROM x26) 6-0, 6-2
Roberta Vinci (ITA x15) bt Petra Cetkovsk (CZE) 6-1, 2-6, 6-2
Angelique Kerber (GER x8) bt Varvara Lepchenko (USA x29) 6-4, 6-7 (3/7), 6-4
Ana Ivanovic (SRB x14) bt Virginie Razzano (FRA) 6-3, 6-2
Carla Surez (ESP x20) bt Mnica Puig (PUR) 6-4, 7-5
Sara Errani (ITA x5) bt Sabine Lisicki (GER x32) 6-0, 6-4
Svetlana Kuznetsova (RUS) bt Bojana Jovanovski (SRB) 6-4, 7-6
Second round:
Stefanie Voegele (SUI) bt Kaia Kanepi (EST) 7-6 (8/6), 3-6, 8-6
Maria Kirilenko (RUS x12) bt Ashleigh Barty (AUS) 6-3, 6-1
Marion Bartoli (FRA x13) bt Mariana Duque (COL) 7-6 (7/5), 7-5
Francesca Schiavone (ITA) bt Kirsten Flipkens (BEL x21) 6-1, 4-6, 6-3
Maria Sharapova (RUS x2) bt Eugnie Bouchard (CAN) 6-2, 6-4