Kazakhstan reached the Davis Cup quarter-finals with a comfortable 3-0 victory over 2019 finalists Canada, while Croatia also went through at the expense of 28-time champions Australia.
Kazakhstan top Group B after wins for Mikhail Kukushkin and Alexander Bublik in the singles, plus Andrey Golubev and Aleksandr Nedovyesov in the doubles.
Croatia defeated Hungary 2-1, meaning Australia finished second in Group D.
Lleyton Hewitt's team cannot qualify as one of the two best runners-up.
Croatia, who won the title in 2018, will face Italy in the first quarter-final on Monday, while Kazakhstan are still waiting to find out which of the runners-up they will play.
Great Britain clinched their spot in the last eight by fighting back to beat the Czech Republic to finish top of Group C.
Leon Smith's team will face either Germany or Serbia, depending on the outcome of Germany's tie against Austria later on Sunday.
Defending champions Spain, without the injured Rafael Nadal, face the Russian Tennis Federation later on Sunday with the winner finishing top of Group A.
The Russians, competing under the name of their federation after the nation was banned from international competition for doping violations, are the favourites to win the event and led by top-five ranked players Daniil Medvedev and Andrey Rublev.
The United States and Colombia also meet to decide who finishes second in Group E behind Italy - and with an outside chance of finishing as one of the best runners-up.
After last year's event was cancelled because of the coronavirus pandemic, the Davis Cup Finals returns for its second edition with a slightly different feel.
The 18-nation finals, self-styled as the 'World Cup of Tennis', was held entirely in Madrid when it launched in 2019, but stretches across three cities - Innsbruck and Turin joining the Spanish capital as a host - this year.
The semi-finals and final will be held in Madrid from 3 to 5 December.