The fourth-seeded Spaniard's flawless performance on Court Philippe Chatrier in Paris against Basilashvili, from the country of Georgia, was his 100th best-of-five-sets match on clay (98-2 record). Nadal improved to 14-0 in third-round matches at Roland Garros.
"I don't know if it was perfection," Nadal, who turns 31 on Saturday, said afterwards. "I went on court thinking it was going to be a tough match. ... I believe I played the best match since a while."
Nadal lost just 12 of his service points -- 22 of 29 on first serve and 19 or 24 on second serve -- and struck 27 winners for victory in 90 minutes.
Nadal will next face fellow Spaniard Roberto Bautista Agut, who needed just over two hours to defeat Czech Republic's Jiri Vesely 6-3, 6-4, 6-3.
"I feel good to reach the fourth round," the No. 17 seed Bautista Agut said. "My returns were really good (and) tactically I think I played well.
"Rafa is playing his best tennis at present on clay. What I will try and do is play my tennis, be aggressive, and I will try and play fast, and then we will see what happens. It is going to be very complicated."
Nadal is now 75-2 at Roland Garros and is five match wins away from 850 career victories (845-180 record).
Fifth-seeded Milos Raonic of Canada advanced to the fourth round when Spain's Guillermo Garcia-Lopez retired from the match with a left thigh injury. Raonic was leading 6-1, 1-0 when Garcia-Lopez stopped.
It was the second retirement of the day as No. 10 seed David Goffin of Belgium was forced out with an injured right ankle against Horacio Zeballos of Argentinia.
Goffin caught his foot under the tarp that was folded at the back of the court while chasing down a ball behind the baseline. Goffin, who was leading 5-4 in the first set at the time, went down immediately and held his ankle before officials helped him off the court.
Spaniard Pablo Carreno Busta, the No. 20 seed, beat 11th-seeded Grigor Dimitrov of Bulgaria 7-5, 6-3,6-4.