Playing in his first tournament since his marriage to long-time girlfriend Kim Sears, Murray led 4-1 in the first set before the Czech Republic's Rosol turned in a furious rally to take the next five games.
The two players then bumped into each other at the changeover which brought complaints from the Scot and an intervention by the ATP supervisor Gerry Armstrong.
Murray later told Rosol: "No-one likes you on the tour, everyone hates you."
The world number three, however, quickly shrugged off the altercation to claim 12 of the next 15 games for the win in exactly two hours.
In the last four the Scot will take on tough Spanish third seed Roberto Bautista Agut, who also had to bounce back after a first set loss to beat Víctor Estrella Burgos of the Dominican Republic 4-6, 6-0, 6-0.
The two other quarter-finals in the second half of the draw see Dominic Thiem take on Gerald Melzer and Philipp Kohlschreiber go up against David
Torrential rain washed out all four quarter-finals on Friday, leaving both the quarter-finals and semis to be played on Saturday.
Murray will seek his second head-to-head victory over Bautista Agut in the semis, having previously triumphed in straight sets in the Wimbledon third round last year.
The Spaniard is in search of his second clay-court title, having won his maiden tour-level title on German soil in Stuttgart in 2014.