The world No.2 sealed Poland's spot in the quarter-finals against the Czech Republic after the world No.12 Badosa was beaten 6-3 6-7 (7-9) 6-1 on the indoor hard-court in Malaga.
Badosa scrapped to get back in the match by winning the second-set tiebreaker, but the former world No.1 roared back with a dominant third set and a break advantage of 8-3.
The match was delayed late in the third set for what organisers called a "medical incident" that apparently took place among the spectators inside Malaga's Palacio de Deportes.
"She pushed me," Swiatek said afterwards. "There was a lot of pressure in the tiebreaker and it went her way, but I knew I could get back in it in the third set."
She underscored the importance of Magda Linette's opening win in the tie for Poland.
"We both did it, and also the whole Polish team," the five-time grand slam winner Swiatek said after making her BJK Finals debut.
Linette took three hours and 50 minutes to finally subdue Sara Sorribes Tormo 7-6 (10-8) 2-6 6-4.
The match had originally been scheduled for Wednesday but was pushed back two days due to heavy rain in southern Spain which caused some flooding in Malaga.
Authorities in Spain are on high alert after massive flash floods claimed more than 200 lives last month.
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