A birdie on the 14th had set Lee up to make history, but pars over the final four holes meant he narrowly missed out.
But the Stirling golfer will still take a two-shot lead over Thomas Bjorn, Victor Dubuisson and Alejandro Canizares into the final round as he sits on 16 under.
Eleven birdies in total ensure Lee is on course to claim his first European Tour title at Crans Montana and he told www.europeantour.com: “[It was] comfortably my best round ever.
“Even as an amateur or in a bounce game, I’ve never managed to get to double figures.
“So this is my best round by far. You just try to focus on each shot you play though. When you hit it and it is executed properly and you see it flying down the pin, it is pretty special.
“For the first 12 or 13 holes that was what was happening. It was pretty surreal to see all these good golf shots coming out in the same day. I’ve been playing quite nicely for a while so it has all come together.”
Lee added: “I would like to have had a heart rate monitor on. From the first I don’t think my heart got under 100bpm.
“There were certain crucial putts that got the heart going and I don’t think it ever slowed down. I’m still not really ready to relax.”
While Lee’s efforts ensured he was in the spotlight, England’s Richard Finch produced a hole in one at the 195-yard par three 13th, his six iron earning him an Omega watch worth £11,750.
The 36-year-old Lee started his third round with six successive birdies, and after a par on the seventh, picked up three more birdies.
Although a par and bogey halted his progress, the Stirling golfer picked up further birdies on the 13th and 14th and history beckoned, ultimately to no avail.
Denmark’s Bjorn, who shared the overnight lead with Scotland’s Stephen Gallacher, picked up three birdies on the front nine before a further two sandwiched a bogey as he finished with 67.
He was joined by Frenchman Dubuisson who produced a mixed round which included three birdies, two eagles and two bogeys including one at the final hole.
Canizares hit an eagle and three birdies on the back nine for a 65 and he said: “I made birdie on the tenth and from there on, everything just happened the right way.
“I’m very happy to finish the way I finished although there were a couple of tough putts in the end, but I still managed to save the round.”
Miguel Angel Jimenez and Tommy Fleetwood both hit rounds of 69 to tie for fifth but they go into the final day three shots behind the trio tied for second.
Ross Fisher had an impressive day, starting with an eagle and going on to hit six birdies for an eight-under 63, but Gallacher was undone by five bogeys for a 72.