The 27-year-old Scot advanced to play Spaniard Roberto Agut, who also initially struggled against Dominican Victor Burgos before winning 4-6 6-0 6-0, later on Saturday after rain washed out play on Friday.
              
With new coach Jonas Bjorkman watching in the stands, Murray forged an early break to race into a 4-1 lead but soon found himself struggling with the Czech's thundering groundstrokes that pinned him to the back of the court.
               
Rosol grew in confidence and reeled off five consecutive games to clinch the first set on his fourth opportunity as the sun briefly broke through the leaden Munich sky.
               
In a repeat of the opening set, Murray again raced to a 4-1 lead but this time remained composed to level the match.
               
The former Wimbledon champion left the Czech no chance in the third set, powering to a 5-1 advantage and sealing his semi-finals berth on the first match point.
               
Germany's Philipp Kohlschreiber takes on Belgium's David Goffin in his last eight game while Dominic Thiem faces Gerald Melzer in an all-Austrian quarter-final.

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