Singles wins Friday by Andrey Golubev and Evgeny Korolev had Kazakhstan on the cusp of clinching the first-round series.
However, Knowle and Peya responded with a 7-6 (7-5), 6-3, 7-6 (7-3) victory over the team of Golybev and Yuri Schukin in the must-win situation.
The teams each had two breaks in the first set but the Austrians didn't see a break point the rest of the way, winning three-quarters of the points on serve in the second and third sets.
The winner of the Austria-Kazakhstan series moves to the Davis Cup quarterfinals against either the Czech Republic or Switzerland. That series was 1-1 after Friday's play.