American Furyk, whose only Major victory came at the 2003 US Open, fired a five-under-par 65 to set the early pace on a mainly sunny day at Oak Hill Country Club, before being caught at the top in the early evening by Masters champion Scott.
Scott surged into contention with a sizzling run of five consecutive birdies from the par-five fourth on a receptive East Course ripe for plundering but his momentum was halted by a weather delay of 70 minutes due to the threat of lightning.
After returning to the course, Australian Scott also birdied the par-four 14th to briefly snatch the outright lead at six under before bogeying the 16th to rejoin Furyk in a tie at the top.
"It was a dream start after kind of a nervous first couple of holes," Scott said of his red-hot outward nine of five-under 30. "Probably the best (birdie) run I've ever had and I just hit really nice shots and didn't leave myself too much work.
"And then after playing so well, I was starting to feel it slip coming in on the last three holes. To make one on 18 and get something out of the round that I felt could have been special was a nice feeling. I did play very well today."
Scott, who led last month's Open Championship by a shot with seven holes to play before a run of four straight bogeys dropped him back into a tie for third, remained level with Furyk by sinking a 15-foot par putt on the final hole.
Furyk, who has not won on the US Tour since his stellar 2010 campaign when he triumphed three times, piled up six birdies and a lone bogey, at the par-four ninth to take early control of the year's final Major.
"I feel very comfortable with what I'm doing with my driver right now, and I've been doing some work on the putting," said the 43-year-old. "Today was probably one of the best putting rounds, if not the best putting round I've had this year."
Englishman Lee Westwood and Canada's David Hearn opened with 66s to share third place, a stroke in front of Americans Robert Garrigus, Matt Kuchar and Scott Piercy, Australians Jason Day and Marcus Fraser, and Paul Casey of England.
Woods appeared to be on track for a one-under round after relying on his renowned scrambling abilities for much of the day but he made a complete hash of the par-four ninth to return a one-over 71.
After hitting his second shot into tangly rough short of the green, he chunked his third under the lip of a bunker from where he splashed out to 15 feet. His bogey putt then lipped out and he had to settle for an ugly six.
"I'm still right there," said four-times champion Woods after also carding two birdies and a bogey. "As of right now, I'm only six back and we have got a long way to go.
"I feel like I played well today and made some nice key putts and the key is I left it in all the good spots too. The round realistically could have been under par easily."
Woods was a heavy favourite coming into this week after romping to a seven-stroke victory at the WGC-Bridgestone Invitational on Sunday, but is hunting his first Major victory since the 2008 US Open.
Defending champion Rory McIlroy, who is seeking to turn his game around after a poor season by his standards, got to three under after four holes on the way to an opening 69.
"I felt like I played really, really well on the front nine," said the Northern Irish world number three, who coasted to victory by a record eight shots at Kiawah Island last year.
"Made a couple of bogeys on the back nine, on 10 and 11, and that sort of halted the momentum. Had a chance on 13, didn't quite make it. But overall it was good."
US Open champion Justin Rose made a solid start with an opening 68 but Phil Mickelson, who won his fifth Major title at last month's Open, battled to a 71 and immediately went back to the range to work on his swing with coach Butch Harmon.
"Now I've got to come out hot tomorrow and get a little more aggressive, attack and try to shoot something in the mid to low 60s to get back in it for the weekend," said Mickelson.
Thirty-five players dipped under par in the opening round on a challenging East Course layout which had previously yielded only 10 72-hole totals under par in five previous Major championships.
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