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.
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.