The United States will face Russia in the other semifinal.
The elimination in front of the home crowd is a big disappointment for the Czechs, who have won the competition — previously known as Fed Cup — six times since 2011. The Swiss are seeking their first trophy.
Earlier, Marketa Vondrousova overcame a slow start on the way to a 6-4, 6-2 victory over Viktorija Golubic to give the Czechs a 1-0 lead.
Bencic started the Swiss comeback by upsetting third-ranked Barbora Krejcikova 7-6 (2), 6-4 to level the series.
The 17th-ranked Bencic broke Krejcikova for a 4-3 lead in the second set before serving the match out.
The Aussies, who reached the final of the last edition of the competition, needed to win the best-of-three series against Belarus to advance from Group B.
Ajla Tomljanovic rallied to defeat Aliaksandra Sasnovich 4-6, 6-2, 6-3 to give Australia an unbeatable 2-0 lead. Storm Sanders beat Yuliya Hatouka 6-3, 6-3 in the opening singles match.
“I was happy I could come out today and the fact that I got the point makes me unbelievably happy,” Tomljanovic said.
For the 131st-ranked Sanders, it was her second win at the tournament. She also beat Belgium’s top player, 18th-ranked Elise Mertens, 3-6, 7-6 (5), 6-0 in her debut for Australia.
“Today, I definitely felt there was more pressure,” said Sanders, who didn’t face a break point. “For me, even if I was a little bit nervous, it was great to have the teammates on the side and they definitely helped with their support.”
The 12 teams in the finals were divided into four groups of three in a new competition format. Only the group winners advance to Friday’s semifinals.
Last year’s edition was canceled because of the coronavirus pandemic.
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