Harriet Dart’s 7-5 6-3 defeat by Anna Karolina Schmiedlova gave Slovakia an unassailable 3-1 lead in Bratislava.
Heather Watson had earlier kept Great Britain in the tie with a 6-0 7-5 win over Rebecca Sramkova.
Slovakia qualify for April’s 12-nation finals in Budapest, while GB will play a qualifying tie to get back to this same stage in next year’s competition.
Britain were without world number 14 Johanna Konta, who opted to sit out the Fed Cup this year to protect her body, and Katie Boulter.
They returned to the World Group for the first time since 1996 by beating Kazakhstan in a play-off last April, and were among eight seeded nations in the qualifiers.
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Dart pays for missed chances
Dart had been beaten by Slovak number one Victoria Kuzmova in a gruelling three-setter on Friday but the 23-year-old Briton showed no sign of fatigue against Schmiedlova, pushing her hard in a tight first set.
But she sent a forehand long to hand Schmiedlova the first set before battling hard at the start of the second, where she carved out two break points on both of the Slovak’s opening two service games.
But in the end it was the failure to convert those chances – missing nine out of 10 break point chances in total – that cost Dart.
Schmiedlova, a former top-30 player who has slipped down the rankings to 199 through injury, showed her pedigree by saving three break points from 0-40 to hold for a 5-2 lead in the second set.
World number 141 Dart did force the Slovak to serve out the match, showing her fighting spirit by saving two match points with two scorching winners, but finally succumbed on the third match point when she sent a backhand long.
“I’m sad to not be able to get a win for the team,” said Dart, who was in tears in her chair at the end of the match.
“I’m proud of the way I was able to fight this weekend. I couldn’t have done anything more. I’m able to hold my head high and take positives from it all.”
Watson makes most of late change to give GB hope
With Great Britain trailing 2-0 overnight, Watson had kept the tie alive with a dominant victory over Rebecca Sramkova.
The British number two had been expecting to face Kuzmova, but the Slovak pulled out shortly beforehand through illness.
Watson said she only found out 55 minutes before the match that she was playing a different opponent.
“I was watching some Netflix to try and get my mind off tennis and relax,” she told BBC Sport. “I found out at five past three that I was playing her so I went straight on to YouTube to start doing my homework as I don’t really know much about her.”
Watson made the most of the late change, punishing world number 202 Sramkova from the off as she won the first six games to take the set in 26 minutes before the Slovak finally held serve in the opening game of the second set.
The 27-year-old Briton, who tightened up her game considerably since Friday’s error-strewn 6-2 6-3 loss to Anna Karolina Schmiedlova, dealt well with the increased pressure in the second set as Sramkova found some form.
World number 74 Watson got a decisive break in the 10th game of the second set when Sramkova went long, with the Slovak then continuing to rack up her groundstroke errors to allow Watson to serve out the match with ease.
But the defeat for Dart afterwards means Great Britain now need to win a play-off tie in April to earn another chance to reach the finals in 2021, while a defeat would send them back to the Europe/Africa Zone.