Kaja Juvan got outsiders Slovenia off to a flyer with a 6-4, 6-1 defeat of a rusty Ajla Tomljanovic.
Former French Open semi-finalist Tamara Zidansek then overcame Daria Saville 6-1 6-4 to give Slovenia an unassailable lead.
"It's really special to share this moment with the team," Zidansek said. "I'm really, really happy with how composed Kaja and I stayed on court today. We both showed a great level."
The Australian pairing of Storm Hunter and Kimberly Birrell did win the doubles rubber against Veronika Erjavec and Ela Milic but they are already up against it in the group.
"I mean, it's a do-or-die, isn't it? It's all or nothing," Australia captain Alicia Molik said of Thursday's clash with Kazakhstan who are the other nation in Group B.
Later on, reigning champions Switzerland suffered a 3-0 defeat at the hands of the Czech Republic.
It always looked like being a tough start for the Swiss against the 11-time winning nation and so it proved.
A gripping battle between two 18-year-olds ended with Linda Noskova edging out Celine Naef 7-6, 4-6, 6-4 after recovering from 4-1 down in the deciding set.
World number 23 Marie Bouzkova then beat Viktorija Golubic 6-4, 6-4 in the second singles to give the Czechs a decisive lead in the Group A clash.
Barbora Krejcikova and Katerina Siniakova rounded off a 3-0 win with a doubles triumph against Golubic and Jil Teichmann.
The Czechs rested Wimbledon champion Marketa Vondrousova after her late arrival from the WTA Finals in Cancun.
The United States are the other team in Group A.