The upset in Perth sent the Kazakhs into the semi-finals of the 18-nation tournament.
In Sydney, world number two Iga Swiatek led Poland into the last eight by winning a rematch of her 2023 French Open final against the Czech Republic's Karolina Muchova.
Sixth-ranked Rybakina earned the first point for Kazakhstan, with the 2022 Wimbledon champion hammering Laura Siegemund 6-3, 6-1.
Zverev then withdrew from his clash against Alexander Shevchenko at the last minute with a bicep strain, leaving the 254th-ranked Daniel Masur to face him.
He put up a brave fight, but Shevchenko was too strong, powering home 6-7 (5/7), 6-2, 6-2 despite a medical timeout in the second set, complaining about difficulty breathing.
"I was struggling in this heat, it was so rough," said the world number 78, who upset Stefanos Tsitsipas in group play on Monday.
"I had a headache and my head was spinning. But I wanted to win this one.
"After (losing) the first set, I knew I had to shorten the rallies and play aggressively, tee off on the ball. I just kept pushing after the first set and I'm really happy about it."
Five-time Grand Slam winner Swiatek scored a battling victory over Muchova, coming out on top 6-3, 6-4 after teammate Hubert Hurkacz was upset in three sets by Tomas Machac.
Swiatek returned to partner Hurkacz and outlast Muchova and Machac in the deciding mixed doubles 7-6 (7/3), 6-3 to keep their tournament alive.
Playing her first event since news broke in late November that she served a one-month ban for a doping violation, Swiatek dropped serve to go 2-1 behind in the first set against Muchova.
But she won the next three games to reassert control and wrapped up the contest comfortably.
"Honestly, I love playing against Karolina because she always brings precision and intensity and a little bit of a different game than most of the girls," Swiatek said.
"So for sure it's a challenge. I'm happy that even though I lost my serve when I thought it's going to be easier, I'm happy that I got my intensity even higher in the last two games to break again."
Swiatek and Muchova had played three times before and they all went three sets, with the Pole winning two including her memorable Roland Garros triumph.
Britain face Australia later in Sydney to determine what the last quarter-final looks like, with the winner playing the already-qualified Italy.
Poland's will meeting the yet-to-be-decided best runner-up from the three Sydney groupings.
The United States, the 2023 winners, take on China later in a quarter-final in Perth.
