Victories over Portugal, Holland and Israel in the pool stages had carried new captain Murray’s team through to a Euro/Africa Zone Group I play-off against the Austrians.
Impressive singles wins for Anne Keothavong and Elena Baltacha swept Britain to victory, setting up the opportunity for them to climb into the second tier of world team tennis.
A play-off in April awaits the British team, who will face one of the losing sides from the World Group II matches.
Keothavong beat Patricia Mayr-Achleitner 7-6 (7/5) 6-3 in the opening match, before world number 57 Baltacha produced an inspired display to crush 48th-ranked Tamira Paszek 6-1 6-4.
Captain Murray, mother of men’s world number four Andy Murray and doubles specialist Jamie, said: “Right now I’m absolutely over the moon that we’ve managed to win all of our matches, to win our group, and then win the play-offs and get ourselves a promotion play-off.
“It is everything that I would have hoped for and I’m just so proud of the team after what they’ve achieved this week.”
Baltacha started the competition superbly, claiming an immediate double break of serve to gain control of her match, and another break early in the second set confirmed her superiority.
The British quartet, which also included the doubles pairing of Laura Robson and Heather Watson, headed to Eilat with captain Murray to attain precisely the goal they have achieved. However, it was far from a given.
Now they have the opportunity to take a significant step for the women’s game in Britain by reaching a stage where the likes of the United States, Japan, France and Australia compete.
Britain will learn their opponents for their April 21-22 play-off when the draw is made on 14 February.
Baltacha, Britain’s highest-ranked woman, won all her singles matches, and Murray said of her fellow Scot: “You’d be hard pushed to find a better number one than Bally.
“She leaves everything out on the court, so it’s exciting times ahead for us with the play-off match, and exciting times ahead for the future because Anne and Bally are playing the best tennis of their lives; the young players Heather and Laura are very exciting prospects and they will learn a lot from being around the older girls.”
Baltacha said: “I went out there, I got so pumped. I thought ‘Right, I’m going to do this,’ and I played so well and was just really pleased that it didn’t have to go to a deciding doubles. I’m never going to forget this day.”
Keothavong said: “Since I made my Fed Cup debut 11 years ago, this is the first time we’ve managed to move up and advance out of this group and I’m just so pleased for the team.”
In other Fed Cup ties, Wimbledon champion Petra Kvitova rallied to beat Germany’s Sabine Lisicki and send defending champions Czech Republic into the semi-finals.
Angelique Kerber later beat Lucie Hradecka 6-4, 6-4 to salvage some pride for the hosts with the 24-year-old German’s first Fed Cup win.
However, Iveta Benesova and Barbora Zahlavova Strycova reaffirmed Czech dominance in the final game of the best-of-five series with a 6-3, 7-6 (4) win over Julia Georges and Anna-Lena Groenefeld in the doubles.
It was the Czechs’ sixth win in seven meetings between the countries and they will next play Ukraine or Italy on April 21-22.
Russia advanced to the World Group semi-finals with a 3-2 win over Spain.
Svetlana Kuznetsova secured the host an insurmountable 3-1 lead over Spain, beating Silvia Soler-Espinosa 6-2, 4-6, 6-3 in the second reverse singles.
Maria Sharapova and Svetlana Kuznetsova easily won their opening singles, but Carla Suarez Navarro narrowed the gap, routing Nadia Petrova 6-0, 6-3 in the first reverse singles on a hardcourt at the Olympic indoor stadium.
The Spanish pair of Nuria Llagostera Vives and Arantxa Parra Santonja set the final score at 3-2 for Russia when, a set down, the Russian duo of Kuznetsova and Petrova retired in the doubles match because of Kuznetsova’s fatigue. Russia will host either Belgium or Serbia in the semi-finals on 21-22 April.