On a day when the United States whitewashed France 3-0, world number four Murray brought Britain level with Italy by beating Potito Starace 7-5, 6-1 after French Open champion Francesca Schiavone had beaten Laura Robson 7-5, 6-3.
However, the Britons were edged out 6-7, 7-6, 10-2 (match tiebreak) in the deciding mixed doubles. Britain now face an uphill struggle if they are to match their run to the final of 12 months ago.
Murray found it hard going in the hot and humid conditions inside the Burswood Dome but after a slow start, he overpowered Starace, breaking once to win the first set and then cruising through the second.
The 16-year-old Robson had been struggling with a cold in the build-up to the match but was impressive against Schiavone, ranked 210 places above her at seven. When Robson broke to lead 5-3 a shock seemed possible but Schiavone won seven straight games to take control and held off a spirited Robson fightback to clinch victory.
Murray and Robson looked on course for victory in the mixed doubles when they broke to lead 4-3 in the second set but as Murray tired, the Italians hit back to level the match and then take the deciding match tiebreak 10-2. Former junior world number one Kristina Mladenovic took the first set against Bethanie Mattek-Sands but that was as good it got for France .
Mattek-Sands won 3-6, 6-3, 6-1 to give the US a 1-0 lead and John Isner made sure there would be no repeat of his epic Wimbledon encounter with Nicolas Mahut as he beat the Frenchman 6-3, 7-6.
It was the first meeting between the two since their 11 hours, five-minute first-round encounter at at the All England Club last June when Isner edged it 70-68 in the final set.
This time, the match was over in just 90 minutes, much to the relief of Isner, who later joined Mattek-Sands to win the mixed doubles 2-6, 6-3, 10-8.
Kazakhstan's Yaroslava Shvedova had to withdraw with a right knee injury. She will be replaced by world number 140 Sesil Karatantcheva.