Kazakhstan stamped their authority over Poland on clay at home, taking full advantage of world number one Iga Swiatek's (21) absence due to injury as Yulia Putintseva (28) stunned Magda Linette (31) 7-5 6-3 before Rybakina doubled their advantage.
The reigning Wimbledon champion fired nine aces and won three out of six breakpoints in her 6-3 6-4 victory over Weronika Falkowska (22) to put Kazakhstan in full control ahead of the final day's action.
The best-of-five qualifiers will feature two singles matches on Friday and two reverse singles matches on Saturday along with a doubles rubber.
"Magda is a rather difficult player so it'll be a difficult match," Rybakina said, looking ahead to her clash with Linette.
"I hope today's match will allow me to feel more confident tomorrow. I'll prepare and hope that I'll be able to win."
France took a 2-0 lead over hosts Britain as world number five Caroline Garcia (29) first battled to a 6-7(2) 7-6(4) 7-6(2) win over Katie Boulter (26) in a marathon lasting three hours 26 minutes.
"Three tie-breaks just says how tight the match was. Katie played really well, I just had to fight and stay in the match," WTA Finals winner Garcia said.
Her compatriot Alize Cornet (33) then beat Harriet Dart (26) 7-6(6) 7-6(3) to extend France's lead.
Czech Republic's former Roland Garros runner-up Marketa Vondrousova (23) beat Ukraine's Marta Kostyuk (20) 6-2 6-1 in the Turkish city of Antalya where the teams are playing due to the war in Ukraine following Russia's invasion of its neighbour.
Krejcikova, the 2021 French Open champion, then eased past Ukraine's Katarina Zavatska (23) 6-4 6-3.
The Czech star, who said she has been supporting the people of Ukraine during the conflict, also pledged her earnings from the qualifiers to the people of Turkey affected by the devastating earthquake in February.
Spain led Mexico 2-0 after Sara Sorribes Tormo (26) thrashed Fernanda Contreras Gomez (25) 6-0 6-0 before Nuria Parrizas-Diaz (31) beat Renata Zarazua (25) 6-4 6-3.
The United States, who have won more titles than any other nation in the competition with 18, took a 2-0 lead over Austria after teenage sensation Coco Gauff (19) beat Julia Grabher (26) 6-1 6-3 and Jessica Pegula (29) secured a 6-0 7-5 win over Sinja Kraus (20).
Canada and Belgium were tied at 1-1 as former U.S. Open finalist Leylah Fernandez (20) first defeated Yanina Wickmayer (33) 6-0 6-3 before Ysaline Bonaventure (28) recovered from a shaky start to beat Rebecca Marino (32) 4-6 6-4 6-4 in two hours 21 minutes.
In the other contests, Italy led Slovakia 2-0, Germany and Brazil were tied at 1-1 and Romania had a 2-0 lead over Slovenia.
The nine winning countries from the qualifiers will join defending champions Switzerland and 2022 runners-up Australia at this year's Finals, which will take place from November 7th-12th.
The 12th nation competing at the finals will be announced once a host has been confirmed.
