The 18-year-old French Open finalist cruised through the first set with four aces, breaking Rybakina in the seventh game, and appeared to have the match firmly in hand.
But Rybakina refused to go down without a fight, fending off four match points in the second-set tiebreak before taking it.
A rattled Gauff struggled in the third set, stumbling into seven double faults as the two players exchanged breaks of serve.
The American then dug deep to find her composure, winning five straight points to clinch the third-set tiebreak before pumping her fist and pointing to her brain to cheers from the crowd.
Poland's world number one Iga Swiatek (21) had no such issues against Ajla Tomljanovic (29) in a largely routine affair on Centre Court.
The Australian qualifier showed a glimmer of life as the two traded breaks in the second set but the twice French Open winner found her footing in the fifth game before wrapping up her 19th straight win on hard courts.
Former world number one Pliskova had no such issues against American Anisimova, breaking her in the fourth game of the first set before running away with the match.
Pliskova fired down seven aces and won nearly three quarters of her first-serve points just days after Anisimova bounced her out in the second round of the Silicon Valley Classic. She next faces Greek powerhouse Maria Sakkari.
Two-time Grand Slam winner Simona Halep (30) of Romania eased past China's Zhang Shuai (33) 6-4 6-2.
