The British No 1 battled back from a set and a break down to triumph 4-6 7-6 (7-5) 6-3 6-0 at a sun-kissed Queen's Club in west London to help Leon Smith's men take an unassailable 3-1 lead in the best-of-five tie.
Murray, who was playing his third rubber of the weekend following Saturday's doubles victory alongside brother Jamie, defied the aches and pains assailing his body to come through.
The Scot's 23rd win from 25 Davis Cup singles matches secured a first victory over old rivals France since 1978 and took Britain back to the last four for the first time since 1981.
It was a herculean three hour and 26 minute effort from the exhausted Murray and, having ended Britain's 77-year wait for a homegrown male winner of the Wimbledon singles title in 2013, he now has his sights on another memorable milestone.
Leon Smith congratulates Murray after a battling display
After wiping away some tears of joy, Murray, who won both singles matches and the doubles with brother Jamie, said: "It feels unbelievable to get through. I used up my last ounces of energy.
"It wasn't looking good in the second set. I was making too many mistakes but I didn't care how I played. I just wanted to win.
"The whole weekend been fantastic. This team has done amazing things. We are punching above our weight."
Britain haven't won the Davis Cup since 1936 and their last final appearance came in 1978, but they have a golden opportunity to end those barren runs this year.
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, having won 12 of his previous 14 meetings.
Simon was in good form on grass however, having made the Queen's Club semi-finals and the Wimbledon quarter-finals before routing James Ward in the opening rubber on Friday.
Having ended Roger Federer's run of 116 unbroken service games at Wimbledon, Simon had no reason to feel intimidated by the Murray.
And despite some rocky moments, the world No 3 and two-time Grand Slam champion sent Britain into dreamland to the delight of the home supporters packed in on Centre Court after completing one of the most tenacious wins of his career.
Lleyton Hewitt and Australia are next for Great Britain
Smith's team will next host Australia in the last four on home soil in September, with a final showdown against Belgium or Argentina awaiting if they see off Lleyton Hewitt and company after they recovered from a 2-0 deficit to win 3-2 against Kazakhstan.
"It should be a great match," Murray added. "Australia have some top young players and Hewitt who has a lot of experience. It will be fun and hopefully there'll be another good atmosphere."
Australia were also the opposition when Britain last won a semi-final in 1978.
