Anne Keothavong began by beating Julia Glushko 6-2 6-1, a result which ensured Britain could not be caught by their rivals in the group.
"I was very pleased to come through against Shahar," said an emotional Baltacha afterwards, "especially to win the second set with six games on the trot was amazing. For me it showed what I am capable of and that’s really exciting. Once I’d won that match point, tears just came straight to my eyes, I had the biggest lump in my throat. But we’ve got one more to go, and we’re going to give it our absolute all.”
Next up for new coach Judy Murray's team is a clash against Austria - if GB win that, they compete in April's World Group II promotion play-offs, and escape the group where they have been stuck since 2005.
"The way the matches have worked, everybody is feeling fresh and lively and we are very much looking forward to facing Austria," Murray added.
"Coming into this competition, I had hoped that we could be in this position, competing in the play-offs, and we’re so close now.”