Even though the US had to make do without the services of their top-four players, Danielle Collins and Shelby Rogers had been expected to steer the 18-times champions to victory considering the highest-ranked player in the Slovakian camp was world No.83 Anna Karolina Schmiedlova.
However, after the teams split the singles matches, the Slovak pairing of Viktoria Kuzmova and Tereza Mihalikova displayed nerves of steel to beat Caroline Dolehide and Coco Vandeweghe in the doubles 6-2 6-7 (5-7) 12-10.
Kuzmova had upset Shelby Rogers 6-4 6-4 while Danielle Collins levelled the tie with a 6-3 6-2 victory over Schmiedlova.
The Americans will have to beat Group C rivals Spain to have any chance of progressing to the semi-finals.
In Group D, Switzerland's Belinda Bencic and Viktorija Golubic won both singles matches on the way to a 3-0 win that eliminated Germany from the tournament in Prague.
Golubic defeated Andrea Petkovic 6-4 7-5 before Bencic, the Tokyo Olympics champion, rallied to overcome Angelique Kerber 5-7 6-2 6-2 to seal the tie.
The Swiss will complete their group stage against the Czech Republic on Thursday.
Earlier on Tuesday, Russia brought Canada to earth by winning 3-0 while 2019 runners-up Australia opened their account by beating Belgium 2-1 in Group B - courtesy of singles wins from Storm Sanders and Daria Gavrilova.
Australia next play Belarus - who Belgium edged 2-1, on Thursday with victory to secure a place in the last four for Alicia Molik's team.
Russia will complete Group A play against France on Wednesday, with Canada eliminated.
The 12 teams in the finals are divided into four groups of three teams. Only the group winners advance to the semi-finals on Friday.
with AP
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