The 33-year-old Furyk, co-leader overnight with Vijay Singh, carded a three-under-par 67 after mixing five birdies with two bogeys at a sun-drenched Olympia Fields.
Woods, bidding this week for a third U.S. Open crown in four years, dropped out of contention with a five-over-par 75 -- his worst score as a professional in eight U.S. Open starts -- after a fan put him off his second shot at the par-five first.
Furyk, who possesses one of the most unorthodox swings in the game, finished at a tournament record 10-under 200 on a day when most of the 68-strong field struggled to maintain any momentum.
One of the straightest hitters in the game, Furyk shattered the previous 54-hole mark of 203, first set by George Burns at Merion in 1981, and later equalled by Tze-Chung Chen at Oakland Hills in 1985 and by Lee Janzen at Baltusrol in 1993.
"I don't think I've sat on a three-shot lead on a Saturday night and I've put myself in a great position for tomorrow," said the American, who has produced 12 top-10 finishes in major championships since turning pro in 1992.
"I got into this position playing the golf course with a certain strategy and I'm going to do the same tomorrow."
Twice major winner Singh, who became the fourth player in U.S. Open history to card a 63 on Friday, bogeyed the last three holes for a 72, slipping back into a tie for third at five-under 205 with Nick Price (69).
Former world number one Price, after a sizzling start featuring five birdies in the first six holes, stumbled round the turn but also managed to birdie the last to keep alive his hopes of a fourth career major title.
"I got off to a great start and I just tried to hang in there," the 46-year-old Zimbabwean said.
"But on the back nine my driving let me down. It was tough out there today."
Australia's Stephen Leaney, who like Furyk birdied the 18th, moved into second place with a gutsy 68 while American journeyman Dicky Pride, whose 66 was the best round of the day, ended up in a share of fifth at four-under 206.
Defending champion Woods, four under overnight after a second-round 66,endured a frustrating day as his title hopes all but disappeared.
"I made nothing out there," said the world number one, who mixed six bogeys with a solitary birdie to finish at one-over 211.
"When you don't make any putts, you can't get momentum. And then I missed a couple of par putts to compound the problem."
Woods had hit a perfect drive at the 576-yard opening hole but then pushed his approach into the right front bunker as the spectator whistled.
"It came on my downswing ... and you can't stop that," he said.
After growling in the direction of the offending fan, he splashed out to 15 feet but was unable to sink the birdie putt.
He dropped his first shot of the day at the par-four fifth, where he hit his approach through the back of the green, another on nine, where he lipped out from six feet, and ran up his third bogey at the 444-yard 10th.
He was never able to recover after that.
Most of the early fireworks came from Price, who had fired a five-under-par 65 on Friday to vault into contention.
He picked up shots on one and two, and then struck a nine-iron approach to 12 feet at the 389-yard third for birdie number three, tying overnight pacesetters Singh and Furyk for the lead.
At the 164-yard fourth, he rifled an eight-iron to six feet, and coolly rolled in the birdie putt to snatch the outright lead.
He collected a two-putt birdie at the 555-yard sixth before running up his first bogey of the day at the 212-yard seventh, where his tee shot plugged in a bunker.
Pride, who began the day at level par after opening rounds of 71 and 69, reeled off four birdies in seven holes on his way to a four-under-par 66.
But the 33-year-old American is happy enough just to be playing at Olympia Fields after being sidelined for four months last year with pancreaitis before having his gall bladder removed.
"It's unbelievable, considering a year and a half ago I was sitting in the hospital (in Orlando, Florida) about to die," he said.