The British pair overcame Sabine Lisicki and Philipp Kohlschreiber 6-3 6-2 in a victory that sees Great Britain move to the top of Group B.
The result means that a win over Russia in Great Britain’s last round-robin match would ensure a place in the final of the Australian Open warm-up event.
Earlier Laura Robson had lost in the singles against Sabine Lisicki 7-6, 6-3 before Murray beat Philipp Kohlschreiber 6-4 6-1 to level the tie at 1-1 prior to the pair’s comfortable mixed doubles victory.
Following the matches, 22-year-old Murray admitted that he had been caught out by the strength of the Perth sun.
“My back is on fire,” he told reporters. “I practised yesterday with my shirt off and I got so, so burnt.
“But I served really well and didn’t really give him too many chances on my serve. I think that was the main difference.”
MORE: The latest football news
MORE: The latest tennis news
BIOGRAPHY: Marcus Rashford