The World number three secured a final showdown against Belgium or Argentina in November with a 7-5 6-3 6-2 win over Bernard Tomic in the first reverse singles rubber today to clinch the best-of-five tie after playing three days in succession at the Emirates Arena in Glasgow.
The 28-year-old's 25th win from 27 Davis Cup singles matches, which took under two hours, secured a first victory over old rivals Australia since 1978.
Britain have now become the first nation to defeat the three other Grand Slam nations in consecutive ties since they beat Australia, USA and France en route to winning the Davis Cup in 1933.
It is a remarkable turnaround for a British team that have risen from being on the verge of relegation to Zone Group III, the lowest level of the competition, in 2010, to being World Group finalists in 2015.