After defeats by Switzerland in their opening ties, both teams needed a win to avoid facing a relegation play-off.
Heather Watson beat Denmark's Karen Barbat 6-0 6-1 in the opening singles before Anne Keothavong lost 6-0 6-2 to world number one Caroline Wozniacki.
Watson and Jocelyn Rae then defeated Wozniacki and Mai Grage 5-7 7-5 7-5.
Britain will play one further tie on Saturday at the 15-team event to decide their seeding for future ties.
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