Wimbledon champion Elena Rybakina had helped Kazakhstan build a 2-0 lead on Friday and the world number seven had no trouble dispatching Magda Linette 6-4 6-2 to give her side an unassailable lead in Astana.
The best-of-five qualifiers featured two singles matches on Friday and two reverse singles plus a doubles rubber on Saturday.
Rybakina's compatriot Yulia Putintseva was scheduled to play in the fourth singles rubber which was not played but she teamed up with Anna Danilina in the doubles.
Poland, without injured world number one Iga Swiatek, finally got on the board as Weronika Falkowska and Alicja Rosolska won 6-3 6-4 in what was essentially a dead rubber.
In Coventry, Britain were no match for France as WTA Finals champion Caroline Garcia ended the home team's challenge by beating Harriet Dart 6-1 6-7(10) 6-1 to give them a 3-0 lead.
After a one-sided opening set which lasted less than half an hour, Dart took encouragement from the home fans in the arena and rallied in the second set tiebreak where she saved two match points.
But the intense second set had taken its toll on Dart, who ran out of gas in the third as Garcia broke early when another double fault from the Briton handed her a 2-0 advantage.
Garcia got the double break and the world number five served out the set to secure France's place in the Finals.
Ukraine were 2-0 down to the Czech Republic after Friday's rubbers and Marta Kostyuk halved the deficit when she came from behind to beat former French Open champion Barbora Krejcikova 3-6 6-1 6-4.
However, Marketa Vondrousova wrapped up the tie for the Czechs with a 6-3 6-4 win over Katarina Zavatska as the teams opted not to play the doubles match.
