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. I'll prepare and hope that I'll be able to win."
France took a 2-0 lead over hosts Britain as world No.5 Caroline Garcia first battled to a 6-7 (7-2) 7-6 (7-4) 7-6 (7-2) win over Katie Boulter in a marathon lasting three hours 26 minutes.
"Three tiebreaks 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 then 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 following Russia's invasion of its neighbour.
Krejcikova, the 2021 French Open champion, then eased past Ukraine's Katarina Zavatska 6-4 6-3.
The 27-year-old, who said she has been supporting the people of Ukraine during the conflict, also pledged her earnings from the qualifiers to the people in Turkey affected by the devastating earthquake in February.
Spain led Mexico 2-0 after Sara Sorribes Tormo thrashed Fernanda Contreras Gomez 6-0 6-0 before Nuria Parrizas Diaz beat Renata Zarazua 6-4 6-3.
In Stuttgart, Germany lost the first match against Brazil but levelled the tie with victory in the second game.
Anna-Lena Friedsam won the first set against Beatriz Haddad Maia but ended up losing 6-3 4-6 3-6.
Tatjana Maria then didn't waste her chance to tie the matchup and defeated Laura Pigossi 6-3 3-6 7-5.
The United States, who have won more titles than any other nation in the competition with 18, face Austria while Italy play Slovakia and Canada take on Belgium.
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.
