World number one Tiger Woods, seeking his 15th major title but a first since the 2008 US Open, was two under after six holes but eventually finished one over after a double-bogey six on the ninth, his final hole.
Lee Westwood took a two-shot lead into the final round of the Open only to card a closing 75 and finish third, a record eighth top-three finish without winning one of the game's four biggest events.
Open champion Phil Mickelson also shot 71 after suffering the same fate - although his last hole was the 18th - while defending champion Rory McIlroy had to settle for a 69 after three birdies in his first four holes and US Open winner Justin Rose bogeyed the last for a 68.
"After the Open I could have looked on it as disappointing but you try and take the positives out of it and carry on," Westwood said after a flawless round with four birdies and no bogeys.
"There was a bit of a backlash last week (at the Bridgestone Invitational) where I struggled to get into it but I managed to get my focus back again and I played very well today. If I can carry on like that hopefully I will have a chance on Monday."
Westwood could easily have been sharing the lead if not for missing a number of decent chances on the back nine, including one from five feet on the 13th when play resumed after a 70-minute weather delay.
"I was a bit unlucky there because when I marked my ball I noticed there was a spike mark in the way and I thought I could do with it raining a lot to flatten it out," Westwood added.
"But when we came back it was still there and my ball hit it and went left, but it was nice to birdie the 17th and 66 with no bogeys is a great way to start."
McIlroy, who has been watching videos of his eight-shot win at Kiawah Island last year for inspiration, said: "I got off to a fast start but the bogeys on the 10th and 11th halted my momentum.
"I came back after the delay with a nice birdie on the 12th but the last four hole stretch is pretty tough. All things considered one under is not a bad score."
Paul Casey had led the English challenge all day until Westwood's score, the 36-year-old in the third group out at 7:35am and posting a three-under 67.
The former world number three, down at 169th in the rankings after struggling for form and fitness before winning the Irish Open in June, carded five birdies and two bogeys; a sharp contrast to 12 months ago when he missed the cut with rounds of 79 and 85.
Casey had started inauspiciously with a bogey on the 10th and followed that with eight straight pars, but then birdied five of his next seven holes and made sure to check out his name high on the leaderboard as he finished.
"It's a while since my name has been up there," said the 36-year-old, who broke his collarbone snowboarding at the end of 2011. "It felt good.
"You don't realise how much you miss something until it's gone and sitting on the sofa watching the big events of the last year and a half I have had lots of extra motivation to get back.
"It added another reason for getting out there and working hard and things like Justin (Rose) winning the US Open is good fuel for the fire.
"I got a bit lucky today with a couple of monster birdie putts from 50ft and 33ft on the fifth and seventh, but we're off and running. I knew I had to get something going after the turn because I had got through the tough part of the course."
Furyk, looking to become the second 43-year-old in succession to win a major after Mickelson's Open triumph, was set to equal the course record of 64 when he reached six under, but bogeyed the last for a five-under 65.
And Scott had dreams of shooting the first round of 62 in major history when he stormed to the turn in 30 with five birdies in succession and reached six under after 14, but bogeyed the 16th.
"There's nothing to complain about with a 65 but I was feeling it when we had the delay and when we came back I had a 233-yard par three with my first swing," Scott said.