The Polish players beat New Zealand’s Michael Venus and Taiwan’s Chan Hao-Ching, 6-2, 6-3, in their second-round encounter on Monday.
Up next for 37-year-old Kubot and 18-year-old Świątek at the tournament Down Under are Australians Astra Sharma and John-Patrick Smith.
Poland’s Świątek earlier on Monday bowed out of the women's singles competition at the Australian Open after being stopped in the fourth round by Estonia’s Anett Kontaveit. The rising Polish tennis star lost 7-6, 5-7, 5-7.
Meanwhile, Poland’s Kubot on Saturday crashed out of the second round of the Australian Open men’s doubles, partnering with Brazil’s Marcelo Melo.
Another Pole, Hubert Hurkacz, was eliminated from the men's singles in Melbourne after losing to Australia’s John Millman, 4-6, 5-7, 3-6, in the second round last week.