The winner-takes-all Group A duel between the two most successful sides in the competition's history was locked at 1-1 after the singles before the Czech pair of Katerina Siniakova and Barbora Krejcikova took the European team through.
The seven-time Grand Slam doubles champions had a little too much craft for Danielle Collins and Taylor Townsend, winning 6-3, 7-5.
Canada will be the Czech Republic's opposition in Saturday's semi-finals.
The US lost to the Czechs at the same stage last year and were intent on setting the record straight, and things looked good when Collins beat Siniakova 6-3, 6-2 in the opening singles.
But Wimbledon champion Marketa Vondrousova, who did not play in the Czechs' win against defending champions Switzerland, levelled things up by crushing Sofia Kenin 6-1, 6-1.
Siniakova and Krejcikova then completed the comeback to keep the Czech team on course for a 12th title.
"This is a team competition and it was great that Marketa came up with the win to make it 1-1," Krejicova said.
"A deciding doubles is always tricky because it's no advantage but I felt that we had a really good team spirit."
Group B was more complex with Kazakhstan needing a 3-0 win to be assured of reaching the last four.
That ambition disappeared as Kaja Juvan thrashed Anna Danilina 6-1, 6-0 before Tamara Zidansek walked on court against Yulia Putintseva with the arithmetic meaning she needed to win one set to send the Slovenians through.
She duly won the opening set 6-2 before Putintseva hit back to level the match, at which point Zidansek retired in the knowledge that her country had done enough to qualify.
"I mean, for me, it's not really sunk in. Once you're here in this moment, it's going fast, so you don't really know what's happening," Zidansek said. "Obviously we still have a lot to do. It's not finished yet."
Kazakhstan won the doubles to seal the tie but it was too late. Slovenia, Kazakhstan and Australia all finished with one win and a 3-3 win-loss record but Slovenia, the lowest-ranked of the 12 nations in the event, had a better sets percentage.
Canada will be the Czech Republic's opposition in Saturday's semi-finals, while Slovenia will face Italy.
