Kazakhstan stamped their authority over Poland on clay at home, taking full advantage of world No.1 Iga Swiatek's absence for the Poles due to injury as Yulia Putintseva stunned Magda Linette 7-5 6-3 before Rybakina doubled their advantage.
Wimbledon champion Rybakina fired nine aces and won three out of six break points in her 6-3 6-4 victory over Weronika Falkowska to put Kazakhstan in full control ahead of the final day's action.
The best-of-five qualifiers feature two singles matches on Friday and two reverse singles on Saturday, plus 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."
France took a 2-0 lead over hosts Britain, with world No.5 Caroline Garcia battling to a 6-7 (2-7) 7-6 (7-4) 7-6 (7-2) win over Katie Boulter in a three hour, 26 minute marathon.
"Three tiebreaks just says how tight the match was,"Â WTA Finals winner Garcia said. "Katie played really well, I just had to fight and stay in the match."
Garcia's compatriot Alize Cornet beat Harriet Dart 7-6 (8-6) 7-6 (7-3) to extend France's lead.
Czech Republic's former Roland Garros runner-up Marketa Vondrousova beat Ukraine's Marta Kostyuk 6-2 6-1 in the Turkish city of Antalya where the teams are playing due to the war in Ukraine.
Krejcikova, the 2021 French Open champion, eased past Katarina Zavatska 6-4 6-3.
The 27-year-old Krejcikova, who said she has been supporting the people of Ukraine during the conflict, pledged her earnings from the qualifiers to the people in Turkey affected by February's devastating earthquake.
Spain led Mexico 2-0 after Sara Sorribes Tormo thrashed Fernanda Contreras Gomez 6-0 6-0 and Nuria Parrizas Diaz beat Renata Zarazua 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 beat Julia Grabher 6-1 6-3 and Jessica Pegula secured a 6-0 7-5 win over Sinja Kraus.
Canada and Belgium were tied at 1-1 as former US Open finalist Leylah Fernandez beat Yanina Wickmayer 6-0 6-3 before Ysaline Bonaventure recovered from a shaky start to see off Rebecca Marino 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 7-12.
The 12th nation competing at the finals will be announced once a host has been confirmed.
- with AP
