No. 3 Djokovic, who is currently tied with Rafael Nadal with 22 major titles, 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 Court Philippe Chatrier.
He attacked Varillas at every opportunity to avoid being drawn into lengthy baseline battles that had seen the 27-year-old win all of 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. Djokovic faces No. 11 Karen Khachanov in the quarterfinals.
A two-time champion in Paris, Djokovic has now reached the quarterfinals for a record 17th time, one more than Nadal, who missed the tournament this year and recently underwent hip surgery.
"I really hope that his rehabilitation process can go well and that we can see him next season. He's so important for our game, on and off the court, one of the greatest legends of tennis in the history of the game," Djokovic said of Nadal. "We want to see a healthy Rafa, no question about it."
Djokovic's 14 straight quarterfinal appearances at Roland Garros also set a record for the most consecutive quarterfinals by a male player in the Open era (since 1968), according to ESPN Stats & Information research. Jimmy Connors reached 13 straight US Open quarterfinals from 1973 to 1985. The second-longest streak at the French Open is nine by Roger Federer from 2005 to 2013.
"I am proud of all the records, but it also means I am not young anymore," said Djokovic, 36. "It was the best level of tennis I have played here, so I am very satisfied."
Djokovic finished with more than twice as many winners, 35-15, and fewer unforced errors. He went 15-for-17 on trips to the net. He put in 80% of his first serves. He converted 6 of 12 break points while dropping his serve only once.
On paper, Musetti had all the tools to unsettle Alcaraz: the touch, a superb backhand and a victory on clay in their only encounter. On court, however, the Italian player had nothing to bother his opponent.
Alcaraz had way too much firepower and grit for Musetti, who also got tortured at the net on Court Philippe Chatrier.
"I had a high quality of shots, I played really aggressive, and he played a complete match from the first point to the last, so I'm happy to be through to the next round," said Alcaraz. "I try not to think about being world No. 1 or the pressure. All these thoughts, I keep them out of my mind. I just want to enjoy myself, entertain the crowd and smile. This is the key to everything."
Although he had been showing some of his best tennis in Paris, Musetti said he was too passive on Sunday, even if he admitted the best player won.
"I was too worried about him. I was always too much in a rush during the game," Musetti said in a news conference. "I think it's a pity because I was playing for sure my best tennis, and today maybe I didn't take the chance that I had."
"The clash we've been all waiting for. I think everyone was expecting it, weren't they? It's here. It's game on," Tsitsipas, who had described Alcaraz as a breath of fresh air to tennis for his charisma and constant smiles, told reporters.
Khachanov showed good resilience to down Italian Lorenzo Sonego 1-6, 6-4, 7-6 (7), 6-1 and reach the French Open quarterfinals for the second time.
The Russian player looked out of sorts in the opening set before finding his range on Court Suzanne Lenglen, using his forehand to devastating effect.
Sonego never recovered from wasting a 4-0 advantage in the third-set tiebreak, which proved to be the turning point. He fell 5-6 behind and saved the first set point but served a double fault on the second.
"After the first set and a half, I was thinking, what am I doing here? He was hitting all over the place, so I decided all I could do was fight," said Khachanov, who made it to the semifinals in the past two Grand Slams at the US Open and Australian Open.
Khachanov is 0-2 in Grand Slams against Djokovic, including a straight-sets loss in 2020, and 1-8 overall, losing their past seven meetings.
"He's one of the toughest tasks, toughest opponents," Khachanov said about Djokovic, "and you cannot count him out."
Sonego was strong on serve, not facing a single break point, as he outplayed his opponent in a one-sided opening set that he sealed when Khachanov netted a crosscourt forehand.
Khachanov brought his best game into the second set to level the contest. Losing the third set broke Sonego's momentum, and his game, while a pumped-up Khachanov skated through the fourth set, prevailing on the first match point.
Tsitsipas continued his charge by battling back from an early break to wrap up the first set and surged through the next, as Ofner's hopes of being the first qualifier to make the quarterfinals of the clay-court major since Marcelo Filippini in 1999 faded fast.
Reuters and The Associated Press contributed to this report.