Poland will now meet Italy who advanced to the semi-final with a 2-1 win over Japan as Jasmine Paolini played a similar starring role for the winners.
Canada will face Great Britain and Australia will play Slovakia later on Sunday for spots in the other semi-final.
Jasmine Paolini was the star for Italy as they beat Japan to advance. (AP PHOTO)
Czech Marie Bouzkova won the opener against Magdalena Frech 6-1 4-6 6-4, and it was up to world No.2 Swiatek to keep Poland's hopes alive.
Undefeated in Billie Jean King Cup in the last five years, Swiatek did just that but was well tested by teenager Linda Noskova, whom she beat 7-6 (7-4) 4-6 7-5 in a tight encounter that lasted two hours and 39 minutes.
Swiatek broke to take a 3-1 lead in the final set before Noskova, who knocked her out of the Australian Open in January, broke back and pulled level at 4-4.
The Pole took the match to even the tie 1-1 before returning on court with Katarzyna Kawa for the decider against doubles world No.1 Katerina Siniakova and Bouzkova.
Swiatek and Kawa then sealed a stunning 6-1 6-4 victory over the Czechs in one hour and 11 minutes.
Five-time Grand Slam winner Swiatek said: "In 20 minutes I'll be dead! It was an exhausting day, but everything worked. We did a great job, we won."
IItaly became the first nation through to the last four after they beat Japan with a deciding straight sets doubles victory by Paolini and Sara Errani over Shuko Aoyama and Eri Hozumi earlier on Saturday.
American-born Japanese player Ena Shibahara, ranked 135th in the world, battled back from a set down to beat 54th-ranked Elisabetta Cocciaretto 3-6 6-4 6-4 in the opening match.
But the tie was locked at 1-1 after French Open and Wimbledon runner-up Paolini beat 56th-ranked Moyuka Uchijima 6-3 6-4 to put the Italians on the board.
Paolini returned for the doubles with Errani, who has five doubles grand slam titles to her name. The pair, who won gold at the Paris Olympics, dominated to beat the Japanese duo 6-3 6-4.
"This year I'm a little bit used to it," the 28-year-old Paolini said when asked about playing singles and doubles back-to-back. "But I will rest now, I will work with the physios here."
Four-times Billie Jean King Cup champions Italy, who were runners-up to Canada in 2023, last won the title in 2013.
