Basilashvili, 25, hit just five winners and committed 34 unforced errors in a one-sided defeat that lasted 90 minutes and finished just as rain clouds were gathering overhead.
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). It was the first time the Spaniard had done so in a completed match at Roland Garros, and he had now played here 77 times-and won 75.
Two more breaks put him in control and a third gave him a spot in the last 16, where he will face Spaniard Albert Ramos-Vinolas who beat France's Lucas Pouille. I think there's ways he can take his game at 30 years old, and older, because hopefully he's going to still want to play for a while. Raonic was leading 6-1, 1-0 when Garcia-Lopez was forced to condede.
The 14-time Grand Slam champion will face compatriot Roberto Bautista Agut in the next round.
In the women's singles event, defending champion Garbine Muguruza of Spain made it through to the fourth round.
The second seed lost it when he was penalised for his behaviour.
"I had a couple of plays in my head, how to play against him, but couldn't find any rhythm", he said.
Two matches ended with retirements.
Djokovic was unhappy for being docked a first serve for taking too long while serving at 4-0 in the fourth set and given a warning for unsportsmanlike conduct later in the same game.
Rohan Bopanna and his partner Pablo Cuevas of Uruguay, however, advanced to the pre-quarters with a hard-fought 5-7, 7-6 6-4 win over the pair of Treat Huey (Philippines) and Denis Istomin (Uzbekistan).
The Belgian 10th seed suffered a freak injury when he tripped over on the court covers, which were situated well behind the baseline, twisting his right ankle in the process and needing to be helped off by two trainers, although his coach later confirmed Goffin had, thankfully, escaped serious injury. "I am really happy because I have a really good feeling".
Williams is a seven-time major champion and was the runner-up at Roland Garros in 2002. I didn't play too numerous five-setters the last couple of years.
Djokovic raced into a 4-0 lead at the start of the fourth set, with his only blip coming when he dropped serve in a fiery fifth game.