The five-time Grand Slam champion and world number two Swiatek fought back to beat Katie Boulter 6-7 (4/7), 6-1, 6-4 and give Poland an unassailable 2-0 lead in the quarter-final in Sydney.
With the Australian Open just over a week away, Swiatek has now won all three of her singles matches at the United Cup to launch her season.
"I am just exhausted," Swiatek, who required a medical timeout in the deciding set and had strapping on one thigh, said after prevailing in just under three hours.
"This match was crazy, so many changes in momentum."
A relieved Swiatek added: "I took four pain killers, a lot hurt, but happy we won."
It was a first-ever meeting between Swiatek and Boulter, who is ranked 24 in the world, but they did practise together before the United Cup.
Swiatek, at her first event since news broke in late November that she served a one-month ban for a doping violation, roared back in the second set after Boulter won a tight first.
The former number one Swiatek took a medical timeout at 2-1 down in the final set as the duo went toe-to-toe in a tense encounter.
Swiatek, 23, finally wrapped it up on first match point when Boulter fired into the net, before running over to the rest of the Poland team courtside.
Before that, the 16th-ranked Hubert Hurkacz, who has not been at his best to start the season and has a new coach, defeated Billy Harris 7-6 (7/3), 7-5 in the men's singles.
The victory against a player ranked 109 places below him gave Poland one foot in the last four.
It was also a badly needed victory on a personal level for Hurkacz, who had lost both his previous singles matches at the United Cup.
"Super-pleased with myself. Today was a really difficult battle with Billy," said Hurkacz. "Definitely happy to get through this one."
The 27-year-old added: "This win is crucial for me to build confidence and get another match in."
Poland will next face a Kazakhstan team led by women's world number six Elena Rybakina after they dumped out reigning champions Germany on Wednesday.
On Friday, Italy face the Czech Republic for a place in the semi-finals against the United States.
The Australian Open, the first Grand Slam of the year, starts in Melbourne on January 12.
