Blinkova, who had already knocked Garcia out in Paris in 2019, prevailed on her ninth match point to end a tussle that lasted nearly three hours.
The world number 56 double-faulted twice on match point as nerves got to her, but she eventually ended it with a powerful serve to set up a meeting with Ukraine's Elina Svitolina.
With Garcia gone, the French women are without a seed in the singles' draw while none of the French men were among the top-ranked.
On a sun-kissed Court Philippe Chatrier, where most of the ticket holders had opted for lunch over tennis, Garcia broke for a 2-1 lead in a strong start to her bid to claim consecutive wins on clay for the first time this season.
She did not face a single break point in the first set and stole Blinkova's serve in the first game of the next set, only for her Russian opponent to break straight back.
Garcia conceded another break in the fourth game with a double fault and a third one in the fifth as Blinkova blitzed through the set despite facing some late resistance.
Blinkova pounced early in the decider and served for the match at 5-4, only for Garcia to save match points to level for 5-5.
She saved other match points at 6-5 after dropping her serve but Blinkova finally got a first serve in to end Garcia's campaign.
