Furyk, the 2003 US Open winner, put himself in contention for a second major title after shooting an even-par 70 to remain at one-under heading into the final day, which has all the makings of a grandstand finish with more than a dozen players within five strokes of the leading pair.
McDowell, who won the 2010 US Open at nearby Pebble Beach, joined the American at the top after capping his impressive round of 68 with a birdie at the 18th at the Olympic Club.
Sweden's Fredrik Jacobsen also shot a 68 to finish alone in third, two shots behind the leaders, with England's Lee Westwood (67), Belgian Nicolas Colsaerts (71), American Blake Adams (70) and South Africa's Ernie Els (68) a shot further back.
John Peterson (72) had a hole-in-one at the 180-yard 13th to join Webb Simpson (68), Jason Dufner (70), Kevin Chappell (68), Australia's John Senden (68) and teenage amateur Beau Hossler (70) at three-over.
Woods was tied for 14th at four-over following his 75. He had started the day tied for the lead at one-under with Furyk and David Toms, who fell to five-over after a 76.
"I'm definitely still in the ball game," said Woods. "I'm only five back and that's certainly doable on this golf course for sure."
Seven different players had a share of the lead at some stage but it was Furyk and McDowell on top at the end of a riveting day's golf in northern California.
Furyk rebounded from two early bogeys to birdie the 268-yard seventh hole when he got up and down from a greenside bunker and then birdied the par-4 11th when he drained a 12-foot putt.
He dropped another shot on the 16th but two-putted the par-5 17th to get back in red numbers.
"Graeme and I are tied for the lead, but there's a bunch of people piled up and close to it," said Furyk.
"But how we play tomorrow at the top of the leaderboard tremendously affects who is still in the tournament and what score needs to be shot to win."
Woods, bidding to win his first major in four years, made a terrible start when he bogeyed four of the first eight holes then finished poorly with bogeys in two of his last three holes.
"I'm just going to have to shoot a good round tomorrow, and post early and see what happens," said Woods. "There's going to be a bunch of guys there with a chance."
McDowell navigated his way through the treacherous first six holes with a string of pars but dropped a shot on the ninth when he hit his approach into the trees.
The Northern Irishman made amends with birdies on the 10th and 13th holes then picked up another shot at the last when he struck his approach to within five feet of the pin.
"There's something about this sea air coming off the Pacific that feels a little bit like home to me," said McDowell. "If this continues, I may have to be getting a little real estate out here or something."
Westwood drained a monster putt on the final hole for his fifth birdie of the round to give himself a real chance of winning his first major after seven top-three finishes during his career.
"Every time you get yourself in contention you learn something new," said Westwood. "I've picked little bits out of all of those, but the main thing is just to go out there and believe that I'm good enough."
Els, who won the US Open twice in the mid 1990s, had the huge galleries roaring with excitement when he chipped in for eagle at the 17th after a terrible start to his round with three bogeys in the first five holes.
"The shot on 17 is what dreams are made of," he said. "I guess it was about a probably almost a 50 yard shot almost into the hole so it's one of those one-in-a-thousand shots."