Davidovich Fokina recovered from a set down to prevail 3-6 6-3 6-4 at Perth's RAC Arena to give Spain a 1-0 lead in the best-of-three-match tie.
Already topping Group A after beating Brazil in their opening fixture, Spain are assured of advancing to the last eight if they topple Poland.
Hurkacz's loss has left women's world No.1 Iga Swiatek needing to defeat Sara Sorribes Tormo in Monday's second rubber to keep Poland's hopes alive.
If Swiatek wins, the tie will be decided in the mixed doubles when the two respective singles combatants from Spain and Poland team up to confront each other.
Casper Ruud has kept Norway's quarter-finals chances alive with a win over Croatian Borna Coric. (Steven Markham/AAP PHOTOS)
In Sydney, three-time grand slam finalist Casper Ruud has kept Norway's quarter-final chances flickering with a 6-4 6-1 drubbing of Croatian Borna Coric.
Donna Vekic had earlier clawed her way to a 7-5 3-6 6-3 victory over unheralded Malene Helgo to get Croatia off to a winning start to their United Cup campaign.
World No.24 Vekic was expected to cruise through the opening singles rubber at Ken Rosewall Arena against the 544th-ranked Helgo.
But Vekic, a quarter-finalist at the Australian Open last year, had to work for two hours and 49 minutes to secure the win.
"I'm glad my serve finally showed up in the third, and made it easier for me," 27-year-old Vekic said.
"I'm happy to get the win. Malene is a really good player and her ranking doesn't do her justice."
Playing their first tie, Croatia are in Group F with the Netherlands, who beat Norway in their opening clash.
After losing to Great Britain in their Group C opener, Team Australia face defending champions the USA in a must-win tie on Monday night in Perth.
France open their Cup quest in the Sydney night session against Germany, who lead Group D after beating Italy 2-1 on Saturday.
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