The 18-year-old came back from a break down twice in the second set to force a tiebreak before pulling off a dramatic comeback to take world number seven Garcia to the decider.
She led by a mini-break in the third-set tiebreak and needed to dig deep to win the final two points, wrapping up her first victory against a top-10 player in two hours and 27 minutes.
Rybakina, making her 2018 debut in St Petersburg, will face the winner of the match between fifth seed Julia Goerges and Italian Roberta Vinci in the quarter-finals.
French Open champion Jelena Ostapenko overcame former world number two Vera Zvonareva of Russia 7-6(6) 6-3 to book a spot in the quarter-finals.
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The world number six withstood a strong start from Zvonareva before edging the opening set after trailing 5-3 and held off a late surge by the 33-year-old to claim the victory in an hour and 39 minutes.
"It wasn't an easy match today... she played really well," the 20-year-old Ostapenko said.
"In the first set I was down a break with her serving for the set, but I'm really glad I came back. I fought for every point."
Twice Wimbledon champion Petra Kvitova kick-started her campaign with a 6-2 6-0 opening-round victory over seventh- seeded Russian Elena Vesnina.
(Reporting by Hardik Vyas in Bengaluru, editing by Ed Osmond)
