The man from Stirling gave a sterling performance at Crans-sur-Sierre, starting his third round with six consecutive birdies and then adding two more before reaching the turn in 28, with his only par on the front nine coming at the seventh hole.
When he picked up a further shot on the 10th, Lee was firmly on course to card the European Tour's first sub-60 round, but a par on the 11th and a bogey at the 12th momentarily halted his charge towards the magical 59.
He responded with back-to-back birdies on the 13th and 14th holes, taking his tally for the round to 11, meaning he only needed to pick up two more shots from his remaining four holes to create history, with the European Tour-record 12 birdies in a round also on the cards.
However, he could not take advantage of the par-5 15th and three more pars followed, as he eventually signed for a career-best round of 61, some 10 shots lower than his opening effort on Thursday.
By the end of his charge up the leaderboard, Lee had moved to 16-under-par 197 and had opened up a two-stroke lead over 2011 champion Thomas Björn of Denmark (67), Spaniard Alejandro Cañizares (65) and Frenchman Victor Dubuisson, who share second place.
"It was comfortably my best round ever," said Lee. "Even as an amateur or in a bounce game, I've never managed to get to double figures. 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 on the same day. I've been playing quite nicely for a while so it has all come together."
Lee, who started the week in 108th position in The Race to Dubai, admitted thoughts of a 59 crept into his mind on the back nine.
He said: "I thought I might get a birdie on 15, but then wondered where would I get the two more birdies I needed as the last three holes are such a tough finish. So in the end it wasn't to be, but then I have to be happy with a 61."
The 36-year-old's aim now is to focus on trying to win a maiden European Tour title. He said: "The last three events I have been eyeing up trying to keep my card for next year and now I find myself in a position where I am trying for my first win."
If Lee should win he will succeed compatriot Richie Ramsay, who last year lifted the gleaming silver trophy.
Dubuisson eagled the seventh and 15th holes en route to his round of 66, but bogeyed the last, while Cañizares had five birdies and an eagle on the 14th hole in his 65.
Björn's round might not have been quite as spectacular, but the Dane knows how to win around Crans-sur-Sierre following his victory two years ago, and his experience could prove vital. "It was a very controlled round," he said. "I just tried to stay out of trouble and take a few chances here and there. I succeeded pretty well in that.
"We saw Craig was nine under through 10 holes. You can't help but see stuff like that. You just want him to shoot 59 when someone gets so close like that. It was a remarkable round because it was tricky today and windy, so all credit to him."
Another former champion, Spaniard Miguel Angel Jiménez (69), is five shots off the pace on 11-under par in a share of fifth position with Englishman Tommy Fleetwood (69), winner of the Johnnie Walker Championship at Gleneagles a fortnight ago.
Overnight leader Steven Gallacher slumped to a one-over-par 72 to be ninth on nine-under par and seven behind Lee, while Paul Lawrie was on 208 after a 70 and Scott Henry on 212 after a 72.