A stroke in front after the rain-delayed second round was completed the previous day, the PGA Tour rookie drained a 25-foot putt to get to 11-under in the third round.
He then narrowly missed a birdie attempt at the par-four fifth to stay three ahead of his compatriot and playing partner Lucas Glover.
Canadian left-hander Mike Weir was a further shot back in third at seven-under after five holes. Only three other players had previously reached double digits under-par at a US Open.
World number one Tiger Woods clinched the 2000 title at Pebble Beach with a 12-under total of 272 and Gil Morgan got to 12-under at the same venue during the third round in 1992.
Jim Furyk reached 11-under at Olympia Fields in 2003 before winning the tournament with an eight-under aggregate. Barnes, who grabbed the overall lead by firing a five-under-par 65 on Saturday for a record 36-hole total of eight-under 132, took advantage early in round three with a birdie at the par-four second.
That put him two strokes clear of the chasing pack before his spectacular eagle at the fourth. While Barnes relished the low-scoring conditions, three-times champion and holder Woods failed to make any headway. Having started the third round a distant 11 strokes off the pace, he bogeyed the par-three 14th, his fifth hole.
Although Woods chipped in from greenside rough to birdie the par-three 17th and get back to three-over, he trailed Barnes by 14 strokes.
Jeev misses cut
Jeev Milkha Singh could not keep up with his past performances and missed the cut after carding a disappointing three-over 73 in the second round.
The Chandigarh golfer had a two-day total of 11-over 151.