The British team trailed by 2-0 after both Watson and teammate and Harriet Dart lost their opening matches on Friday.
However Watson impressed in her second match against Slovakia's Rebecca Sramkova.
She raced into a 6-0 lead in the first set to give fans confidence that a routine victory was on the cards, but it ended up anything but easy.
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Stramcova fought back in the second set to push Watson all the way, with the scores balanced at 5-5 before Watson claimed two games to win a point, leaving the tie at 2-1 before Dart's next match.
That left Dart to take on Anna Schmeidlova, a former top-30 player.
While the Briton fought bravely she lost 7-5 6-3 in straight sets as Slovakia sealed their passage to April's event. Great Britain will have a play-off in April.
Earlier in the day, Spain completed a 3-1 win over Japan to secure their passage to the Fed Cup finas, when Carla Suarez Navarro completed one of her final career matches with a win over Kurumi Nara, whom she beat 6-1, 6-3.
In the doubles, Shuko Aoyama and Ena Shibahara defeated Lara Arruabarrena and Aliona Bolsova Zadoinov 6-2, 6-3.
Elsewhere, the US lead Latvia 2-0 and the Netherlands and Belarus will play a decisive doubles match.
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