The British number one battled back from a set down to triumph 4-6 7-6 (7-5) 6-3 6-0 at Queen’s Club in London to help Leon Smith’s men take an unassailable 3-1 lead in the best-of-five tie.
Murray looked to be on his last legs after his Wimbledon semi-final defeat to Roger Federer just over a week ago, but he had never lost to Simon heading into their latest clash.
Despite some rocky moments, the world number three sent Britain into the final four for the first time since 1981 to the delight of the home supporters packed in on Centre Court.
Nine-time winners France, who finished as runners-up to Switzerland last year, suffered their first defeat against Britain since 1978.
Smith’s team will now meet Australia in a mouthwatering home semi-final on September 18-20 at a venue yet to be decided.