The Group D decider pitted arguably the two strongest teams against each other, and victories for Marketa Vondrousova and Katerina Siniakova booked the Czechs a last-four spot on Saturday.
The remarkable strength in depth of women's tennis in the country has earned them six of the last 10 titles, and they will fancy their chances of adding to that despite the withdrawal through injury earlier this week of leading singles and doubles player Barbora Krejcikova.
Captain Petr Pala changed his line-up around completely from their victory over Poland and it paid dividends, with former French Open finalist Vondrousova, who led her country to victory over Great Britain in April before being sidelined by injury, defeating Danielle Collins 6-3 6-3.
Siniakova, better known as a doubles player, then took the number one role against Gauff, the highest-ranked competitor at Glasgow's Emirates Arena.
The 18-year-old, who headed to Scotland straight from the WTA Finals, was unable to find her best form as she slipped to a 7-6 (1) 6-1 loss, sealing victory for the Czechs.
Gauff, who lost all her matches on her debut at the WTA Finals, said: "She definitely played well today from the last times I played her but, to be honest, I didn't play my best tennis.
"We had a great team. I think overall this week was positive. I needed a week like this. Obviously not the result we wanted. We came to win. I think, for the next time, we will do better."
Earlier, victories for Belinda Bencic and Viktorija Golubic propelled Switzerland past Canada and into the last four.
Both teams had comfortably dispatched Italy in their opening matches, making it a head-to-head fight for the last four.
Golubic set the Swiss, last year's beaten finalists, on their way by coming from a set down to upset former US Open champion Bianca Andreescu 2-6 6-3 6-4.
Bencic then came out on top of an intriguing tussle between the number one players, beating last year's US Open runner-up Leylah Fernandez 6-0 7-5.