The German 28th seed got the better of Murray in the first set, winning it 6-1. However the Scot took control from the second set, seeing him off with 6-1 in the second and third sets.
In doing so, he has become the first British player to make the final four in the competition in almost 80 years.
The fourth seed is next expected to face Novak Djokovic, assuming the Serb is victorious against Robin Soderling this evening. Djokovic, undefeated so far this year, beat Murray in the Australian Open final earlier this year.