The fourth seed needed only 90 minutes to see off the Georgian and secured his place in the fourth round with his most one-sided win at Roland Garros.
Nadal, 31 on Saturday, has now won 98 of 100 best-of-five clay-court matches.
The Spaniard is bidding to become the first player in the Open era - and only the second in history - to win 10 titles at any Grand Slam event.
Margaret Court is the only player to have won 10 or more titles at one Grand Slam event, winning the Australian Open on 11 occasions between 1960 and 1973.
Nadal has a 75-2 win-loss record at Roland Garros with his only defeats coming against Robin Soderling in the fourth round in 2009 and Novak Djokovic in the 2015 quarter-finals.
This was the first time in a completed match at the tournament that Nadal has only lost one game.
Reigning champion Djokovic faces Argentine Diego Schwartzman later.
Nadal won the first set in 23 minutes and won the next five games before world number 63 Basilashvili finally got on the scoreboard.
The Spaniard wrapped up the set by winning the next game and, with a storm threatening to interrupt play, clinched victory in clinical fashion with his 27th winner.
The 14-time Grand Slam champion will face compatriot Roberto Bautista Agut in the next round.
My best match in a while - Nadal
"It is always important to be through, that's the most important thing," said Nadal. "But obviously when you have positive feelings it is even more important.
"Basilashvili had been playing well. He won against Gilles Simon in the first round who is a tough opponent and also Viktor Troicki.
"I'm happy because I had never played against him and I knew it would be tough.
"He hits the ball so quick but I believe I played my best match in a while."
Contrasting fortunes for Goffin and Ra
Belgium's David Goffin retired with an ankle injury after slipping while leading in the first set of his third-round match against Horacio Zeballos.
The 10th seed, who was 5-4 up on Court Suzanne Lenglen, was chasing a deep backhand and slid into a cover, hurting his right ankle.
World number 65 Zeballos will now play Austrian sixth seed Dominic Thiem.
Thiem, 23, beat American 25th seed Steve Johnson, who has been "an emotional mess" after the recent death of his father, in an enthralling 6-1 7-6 (7-4) 6-3 victory.
Thiem is considered one of the outside favourites to win the tournament after beating favourite Rafael Nadal in straight sets in Rome earlier this month.
Fifth seed Milos Raonic reached the last 16 after Guillermo Garcia-Lopez retired because of a thigh injury.
Raonic won the first set 6-1 in 21 minutes and led 1-0 in the second when the Spaniard retired at the changeover.
The Canadian will now face Pablo Carreno Busta who beat Bulgarian 11th seed Grigor Dimitrov 7-5 6-3 6-4.