The world number 94 had never won a match at a Slam before reeling off three successive five-set victories at Roland Garros to tee up a first-ever meeting with 22-time major champion Djokovic.
Varillas is the first player from Peru to reach the second week of a Grand Slam since Jaime Yzaga made the US Open quarter-finals 29 years ago.
Djokovic is bidding to reach the French Open last eight for a 14th consecutive year and is yet to drop a set in the 2023 tournament, but Varillas is not fazed.
"When you enter the court the options are 50-50, it's one against one," said Varillas after his dramatic victory over Polish 13th seed Hubert Hurkacz on Friday. "I'll try to go and win."
He has spent the majority of his career on the lower circuits, winning five ITF Futures titles and five second-tier Challenger events.
Varillas' parents are in Paris watching him play in a European tournament for the first time and he credits them for his belated success.
"My parents took me to these summer camps where I played a lot of sports," he said.
"I loved tennis and soccer, and I kept playing them until I was 15 or 16. I was better at tennis, so I continued, and here I am."
Varillas had made just over $800,000 in career prize money before arriving at Roland Garros but is already guaranteed another 240,000 euros ($257,000) even if he loses to Djokovic on Court Philippe Chatrier.
He would add another 160,000 euros to that tally if he pulls off one of the biggest shocks in recent French Open history.
No player has ever won four consecutive five-set matches at a Grand Slam tournament.
Varillas, who turned pro 10 years ago, hopes his exploits encourage youngsters back home to target a professional tennis career.
"It is something very positive for my country, where there are not so many professional players," he added.
"If you work with discipline and perseverance, and if you believe in yourself, you can dedicate yourself professionally to this.
"I think it can be a good push for the children to pursue their dream of being tennis professionals."
Varillas would climb into the world's top 50 with victory.
