The class of world number two Murray told in the third set as he eased away to a 6-4 3-6 6-1 win in the first singles quarter-final of an ATP tour event to feature two British men since Tim Henman beat Greg Rusedski in Adelaide in 2002.
Yet Murray had seen enough here the 21-year-old who helped Britain win the Davis Cup last season to tell the crowd during an on-court interview afterwards: "Obviously Kyle's the future of the game in this country.
"It's important that he's given time to develop at his own rate and that too much pressure isn't put on him. If he's given time and a little bit of space to keep developing, he can go very far."
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Edmund, the world number 85 who says Murray has been the inspiration in his development, produced a tremendous second set to illustrate his promise but Murray moved up another gear to continue his assault on a record fifth Queen's Club title.
With Wimbledon just over a week away, Murray had one scare when he slipped on the wet grass and fell awkwardly but reported afterwards: "It was a bit of a shock at first but I'm fine."
Murray now faces Croatia's former Queen's winner Marin Cilic, who beat American Steve Johnson 6-7(3) 6-3 6-4.
Milos Raonic's new coaching link-up with John McEnroe took another smooth step forwards as the Canadian overpowered Spain's Roberto Bautista-Agut 6-1 6-4 to set up the other semi against Bernard Tomic, who beat Gilles Muller 7-6(5) 4-6 6-2.
(Reporting by Ian Chadband; Editing by Toby Chopra and Ken Ferris)
