The world number three handed his side an unassailable 3-1 lead in their World Group semi-final clash against Australia following his 7-5, 6-3, 6-2 victory over Bernard Tomic in the first reverse singles rubber on Sunday.
Britain's Dan Evans lost 7-5 6-4 to Thanasi Kokkinakis in the final rubber.
Britain will now face an away trip to take on Belgium in the final, due to played between November 27-29, after they defeated Argentina 3-2 in the second semi-final tie.
But the 28-year-old revealed that a back injury sustained in training last Tuesday had been a cause for concern throughout the tie.
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"My back had been giving me a lot of trouble this weekend and for a few days before the tie as well," Murray said.
"The previous issues I've had with my back have been completely different. My back was fine during the US Open and all through that stretch.
"I didn't feel it when I was playing on the Tuesday so I don't know exactly what happened.
"Sometimes after you have played a lot of tennis and then you do take a break the muscles and everything stiffen up and you can have some issues."
- first title since 1936 -
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It puts nine-times winners Britain, on the verge of relegation to Zone Group III five years ago, a step closer to an historic first title triumph since 1936.
"Obviously I'm delighted to get through. We knew this would be difficult, Australia have so much depth and experience but we fought extremely hard all weekend," the world number three said.