Spain's Nadal, who is bidding to win his 10th Roland Garros title, was in top perform and took just 90 minutes to send Georgia's Basilashvili packing.
Earlier, Garbine Muguruza clinched a last 16 berth with a 7-5, 6-2 victory over Yulia Putintseva, while men's tenth-seed David Goffin retired due to an ankle injury when leading 5-4 in the first set in the third round.
The fourth-seeded and the reigning champion outscored Putintseva 26-14 in winners, and reached the 73 per cent in winning on first serve, 24 percent higher than Putintseva, reports Xinhua news agency.
The Spaniard broke the serve in the first game, before her 22-year-old opponent fought back to build a 3-1 lead.
After the two players levelled the score at 5-all, the fourth-seed regrouped herself to take away the first set at 7-5 on Putintseva's forehand forced error.
Muguruza met less resistance in the second set on her way to last 16. After colliding with an advertisement plate, Putintseva showed her frustration by kicking the plate when she lost the point. Muguruza sealed the win at 6-2.
Muguruza will fight for a quarter-final spot with local favorite Kristina Mladenovic or Shelby Rogers of the United States.
In a game lasting just less than one hour, Goffin and Horacio Zeballos held their serve twice for a 2-2 tie, before Goffin's first break came in the fifth game.
However, after leading 5-4, the Belgian sprained his ankle in an attempt to hit the ball. He called for a medical treatment, before finally making a decision to retire from the match.