Jamieson, who made the cut by one shot, reeled off 11 birdies and seven pars at the Oceanico Victoria course in Vilamoura to finish with a 14-under tally of 199.
His 60 was the 18th on the tour where no one has ever returned a 'magical 59'.
Jamieson is two strokes behind fellow Briton Paul Waring who shot a four-under-par 67, along with Ireland's Simon Thornton, Jamie Donaldson of Wales and South African Hennie Otto.
The 29-year-old Scot narrowly failed with his attempt at a record-breaking birdie at the 18th when his chip just missed the cup.
"I thought it was going in but unfortunately it hung on the lip and went past the hole," Jamieson told European Tour Radio.
"But the fact that I gave myself a chance there at the last hole, I couldn't ask for any more."
Jamieson has only one tour victory to his name and that came at the Nelson Mandela Championship in South Africa in December.
Following his round, he predicted that the perfect conditions in the Algarve would see him overtaken at the top of the leaderboard, and so it proved, with England's overnight leader Waring hitting four birdies on the back nine and five to lie at 16-under par.
"I started a bit poorly and didn't quite hit it close enough but holed some nice putts towards the end of the day," said Waring, who is playing on a medical exemption after an operation on a wrist injury and is yet to win on the European Tour.
"I feel pretty good, two ahead, so we will see what happens tomorrow."
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