The four-time holder of the Coupe Suzanne Lenglen displayed great resilience to win 1-6, 6-3, 7-5 in in two hours and 29 minutes on Court Philippe Chatrier.
Swiatek will next face Elina Svitolina for a spot in the semi-final, after the Ukrainian earlier saved three match points to eliminate last year's runner-up Jasmine Paolini.
After winning Roland Garros last year to complete a remarkable treble of clay-court victories in Madrid, Rome and Paris, the Pole's turbulent 2025 looked set to hit a new nadir as she found herself a set and a break down against the 12th-seeded Kazakh.
Following early exits at the Madrid and Italian Opens this season, Swiatek appeared to be headed for a similarly disappointing result at the tournament she has dominated since her first title in 2020.
"Well it was tough you know, first set I felt like I was playing against Jannik Sinner", said Swiatek.
"I needed to do something to get back in the game, but with her playing like that I didn't feel like I had much hope."
Rybakina hardly put a foot wrong in the first set as she raced into a 5-0 lead, with fifth seed Swiatek finally getting on the scoreboard after a lengthy service game before the world number 11 served out.
But the 24-year-old showed her fighting spirit to level up the second set after being broken in the first game by Rybakina.
She racked up double-faults on crucial points in the fifth game, but managed to secure a vital hold, which turned the tide of the match.
The five-time major winner then broke to love in the very next game, before forcing a decider.
Swiatek seized the advantage in the seventh game of the third set, but another untimely double-fault by the Pole in the next game restored parity.
At 5-5, Swiatek made her move as she broke Rybakina to 15, before holding for a gritty win.
Swiatek now holds a 5-4 head-to-head record over Rybakina, after claiming her first victory against the 25-year-old on the red dirt.