Kazakhstan into Billie Jean King Cup Finals with France and Spain
Rybakina won 6-4, 6-2 in the third rubber in Astana having also won Friday against Weronika Falkowska when teammate Yulia Putintseva defeated Linette.
"I'm delighted that Kazakhstan will be able to fly our flag on this big stage once again," said Rybakina after the tie against a Polish side missing injured world number one Iga Swiatek.
World number five Caroline Garcia gave France the winning point with a 6-1, 6-7 (10/12), 6-1 win over Britain's Harriet Dart in Coventry.
Britain, who reached the semi-finals last year, were missing Emma Raducanu, leaving 117th-ranked Dart as their top singles player.
On Friday, Garcia and Alize Cornet had to play almost six hours between them to edge out Katie Boulter 6-7 (2/7), 7-6 (7/4), 7-6 (7/2) and Dart 7-6 (8/6), 7-6 (7/3) respectively.
"She (Garcia) knew the job was not done yet, yesterday evening she told me 'I'm ready to go tomorrow'," said France captain Julien Benneteau.
Despite dropping serve once, Garcia broke four times in the first set.
Dart fought back in the second with two double faults costing Garcia on two match points.
But she got back on track in the third set, breaking at the start and then again in the sixth game to win.
Pegula, Gauff give US advantage
On clay in Marbella, Nuria Parrizas Diaz, ranked 80, sealed victory over Mexico for hosts Spain, 6-3, 6-0 against 196th-ranked Marcela Zacarias.
Parrizas Diaz also won her singles tie in straight sets on Friday, along with Sara Sorribes Tormo, to ensure Spain's presence in the finals in November at a venue to be decided.
The Czech Republic saw off Ukraine, with Marketa Vondrousova sealing the winning point against Katarina Zavatska 6-3, 6-4, after former French Open champion Barbora Krejcikova lost 3-6, 6-1, 6-4 to 38th-ranked Marta Kostyuk.
On Friday, 12th-ranked Krejcikova eased past Zavatska after 2019 Roland Garros runner-up Vondrousova dispatched Kostyuk.
The matches were played in the Turkish city of Antalya because of the war in Ukraine.
In Stuttgart, Germany's Anna-Lena Friedsam made up for losing the hosts' opening rubber by clinching the winning point with a 6-1, 6-0 thrashing of Brazil's Laura Pigossi.
Friedsam, ranked 91, lost in three sets to world number 14 Beatriz Haddad Maia on Friday before Tatjana Maria saved a match point to haul Germany level against Pigossi.
Jule Niemeier earlier defeated Haddad Maia 7-6 (7/3), 3-6, 6-2.
Record 18-time champions the United States, led by top-10 players Jessica Pegula and Coco Gauff, are 2-0 up against Austria in Delray Beach.
On Friday, Gauff beat Julia Grabher 6-1, 6-3, before world number three Pegula defeated Sinja Kraus 6-0, 7-5.
Defending champions Switzerland and 2022 runners-up Australia received automatic qualification to the November 7-12 finals, which will feature four groups of three.
