The sixth-ranked Noskova, who claimed her first grand slam title at the All England Club this year, served an ace on her first match point to beat Katie Boulter, Alex de Minaur's wife, 6-2, 7-6 (7-3).
That sealed the Czechs' win over Britain, who had put out Australia in the qualifiers in April.
In all, Noskova produced 15 aces to Boulter's four.
Earlier, Bouzkova beat Sonay Kartal 7-6 (7-2), 4-6, 6-4 in a tie lasting a minute short of three hours.
Karolina Muchova, who was beaten by Noskova in the all-Czech Wimbledon final, was slated to play a decisive doubles match with partner Katerina Siniakova, the doubles world No.1, but that wasn't necessary in the best-of-three series.
In the semi-finals the Czechs will face either Spain or Kazakhstan, who play Wednesday.
The other quarter-finals feature two-time defending champions Italy vs China and Ukraine vs Belgium.
In the draw for the men's Davis Cup Final 8, South Korea will play three-time defending champions Italy after becoming the first Asian nation to qualify since the new format was brought in in 2020.
Spain, the 2025 runners-up, drew Austria.
The Czechs, who eliminated the 32-times champion United States in the final qualifying round, will face 2022 champion Canada. Alexander Zverev's Germany faces Britain.
Australia were knocked out in the first round by Ecuador, who later lost to Britain.
The Final 8 is scheduled for Bologna, Italy, in late November.
