Badosa, ranked 13th in the world, won 6-2 3-6 6-4 over Rybakina, giving Spain a chance to go into the semi-finals if they win over Britain on Thursday.
"It was my first time and I think I enjoyed every minute on court," Badosa said.
In a late Group D match on Wednesday, Coco Gauff (18) and Caty McNally(20) beat Magda Linette (30) and Alicja Rosolska (36) 6-1 6-2 to earn the United States a 2-1 win over Poland, who are without world number one Iga Swiatek.
The teams were level before the decisive doubles match after Danielle Collins' (28) opening victory was canceled out by Linette's defeat of Madison Keys (27).
McNally said team captain Kathy Rinaldi had told them to enjoy the occasion.
"Kathy kept reminding us to have fun. That's the most important thing for both us, just to enjoy it," she added.
Earlier in the Billie Jean King Cup, Belinda Bencic (25) led Switzerland to victory over Italy in their opening tie, while Slovakia bounced back from a tough loss to Australia to defeat Belgium and get their campaign back on track.
The competition, formerly called the Fed Cup, was renamed in honour of the 12-time Grand Slam champion and restructured into a format concluding with 12 nations competing over one week for the title.
