The Serbian, who is currently tied with Spain's Rafael Nadal (37) at 22 majors, was untroubled by his 94th-ranked opponent, the first Peruvian in 29 years to reach a Grand Slam fourth round, on a windy day on Philippe Chatrier court.
He attacked Varillas at every opportunity to avoid being drawn into lengthy baseline battles that had seen Varillas win all his three previous rounds in five-set marathons.
Varillas lost the vast majority of the shorter rallies and could not find a way back into the match as Djokovic set up a quarter-final against 11th seed Karen Khachanov (27).
A two-time champion in Paris, Djokovic has now reached the quarter-finals at the French Open for a record 17th time, one more than 14-time champion Rafa Nadal who is absent this year due to injury.
"I am proud of all the records but it also means I am not young any more," said Djokovic after the win."It was the best level of tennis I have played here so I am very satisfied."
"I put in a lot of effort like all the players and the situation today is not the same like, say, 10 years ago. When I play like today I am satisfied and motivated to continue."
