Nadal is back on the Philippe Chatrier court and he takes on the home favourite Gasquet, the No 27 seed, who has never beaten the world No 1 in 15 competitive settings.
Petkovic had pushed her highly fancied opponent close in the first set with the Romanian eventually only winning it by a 7-5 scoreline, but the German then suffered an injury in the second set which required the trainer to treat it several times.
Rafael Nadal has recorded another dominant win at the French Open, with the Spaniard seeing off Richard Gasquet to reach the fourth round.
Two-time Grand Slam champion Angelique Kerber reached the quarterfinals of the French Open for the second time by beating Caroline Garcia 6-2, 6-3. "I really have trouble with his forehand that comes to my shoulder every time".
Del Potro next plays either ninth-seeded John Isner, the last American left in the men's singles after eliminating the last Frenchman standing.
She was unable to recover from this, meaning Halep dominated and won the second and decisive set by a convincing 6-0 scoreline.
Next up for Williams as she competes in a major tournament for the first time in 16 months - and since giving birth in September - will be a familiar foe: Maria Sharapova.
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Halep has twice lost in the final at Roland Garros, including past year.
Del Potro had eight aces and broke his Spanish opponent's serve six times on Court Philippe Chatrier, taking just 29 minutes to win the third set.
Gasquet actually won their last encounter - but it was at the Kooyong Classic exhibition event this year. He will face No. 18 Fabio Fognini next. The Belgian will take on Italy's Marco Cecchinato for a place in the quarter-finals.
He will next face 11th seed Diego Schwartzman after the diminutive Argentine underlined his clay court capability with a 7-5 6-3 6-3 victory against the unseeded Borna Coric.
The 64th-ranked Mischa lost in the third round to 2017 US Open runner-up Kevin Anderson 6-1, 6-7 (3), 6-3, 7-6 (4) on Saturday. Cilic reached the final at Wimbledon past year and at the Australian Open in January. He'll next face No. 11 seed Diego Schwartzman, a 7-5, 6-3, 6-3 victor against Borna Coric.
In the completion of two matches from Friday, Karen Khachanov finished off Lucas Pouille and David Goffin saved four match points to put an end to Gael Monfils in a fun five setter 6-7 (8), 6-3, 4-6, 7-5, 6-3.
Third-seeded Garbine Muguruza of Spain has crushed Samantha Stosur 6-0, 6-2 to advance to the fourth round of the French Open.