The Americans will take on Britain, who are back in the final four of the women's team competition for a second straight year after defeating Japan in Shenzhen.
Navarro survived a slew of drop shots from Yulia Putintseva and had to save two match points before she completed a 7-5, 2-6, 7-6(6) victory to get the Americans on the board in their quarter-final tie.
Former Wimbledon champion Elena Rybakina levelled things up for Kazakhstan with a clinical 6-4, 6-1 win over world number seven Jessica Pegula.
The American quickly hit the reset button and stepped right back on court to partner Taylor Townsend for a 6-2, 7-6(1) win over Rybakina and Putintseva in the deciding doubles clash.
"I'm so happy and so proud of all the ladies on this team. We knew it was going to be so tough when we saw the draw even a couple months ago," said USA captain Lindsay Davenport.
"We didn't come here to get to the semis. This is the first step in what we knew was going to be a tough week."
Navarro was the hero of the day as she battled Putintseva for two and a half hours and erased deficits in all three sets she contested, including a 4-6 hole in the deciding tiebreak.
"Just a lot of grittiness and a lot of resilience and the pride of playing with my country's flag on my shirt," said Navarro of the key to her victory.
In the second quarter-final of the day, world number 82 Sonay Kartal got Britain off to a successful start with a 6-3, 7-6(4) win over Japan's Ena Shibahara.
Kartal, 23, made her BJK Cup debut at the qualifiers in April this year, winning both matches she contested and remains undefeated in the competition.
Katie Boulter finished the job for the Brits with a 6-2, 6-1 victory over Moyuka Uchijima in 68 minutes.
"Sonay put a clinical performance out there so she makes it really easy to follow," said Boulter.
Britain narrowly fell 2-1 to Slovakia in last year's semi-finals and Boulter is hoping there will be no repeat against the USA on Saturday.
"We've got another shot at the final. Hopefully we can keep going further. I hope this is our time now," she said.