The 27-year-old beat Jerzy Janowicz 6-2, 6-4 in Perth to level the clash after partner Heather Watson was brushed aside 6-3, 6-1 by world No5 Agnieszka Radwanska.
The tie hinged on the mixed doubles which the British duo lost 6-4, 6-4 as Group B leaders Poland won 2-1 overall to maintain their unbeaten record and assume pole position for Saturday’s final.
Poland will confirm their place should they beat France on Friday, while Britain play hosts Australia in the knowledge only the group winners will make the finale.
Murray reeled off nine successive games in beating Janowicz and confirmed the shoulder problem picked up while defeating Rafael Nadal at the Mubadala World Tennis Championship last week is easing.
“I moved well and served good and got a lot of returns back in play so I was happy,” he said. “My shoulder felt much better today. As I’m playing the matches, it’s starting to feel a little bit better.”