Andreeva downed Moscow-born Gracheva 7-5, 6-2 at Roland Garros to book a last-eight spot at a Grand Slam for the first time.
Aged 17 and 27 days, she is the youngest female player to make the quarters since Bulgarian Sesil Karatantcheva achieved the feat at the age of 15. Andreeva, who has only dropped one set in the tournament so far, will next face a daunting task up against world number two Aryna Sabalenka.
Gracheva, who took French nationality last year after qualifying under residency rules, was the last home player left in either singles draw.
"When they were screaming 'Varvara, Varvara, Varvara', I imagined they were screaming my name and it really helped," Andreeva said.
"It was tough to play against a French crowd but I'm happy that I managed to get through it."
She made the third round on her Grand Slam debut in Paris 12 months ago and has since reached the last 16 at both Wimbledon and the Australian Open.
Andreeva is the only unseeded women's player left in the tournament.