World number 13 Scott conjured up eight birdies in benign conditions at Lytham to finish on six under par, one ahead of Scotland's Paul Lawrie, American Zach Johnson and Belgian Nicolas Colsaerts.
Former world number one Woods played beautifully controlled golf to birdie the first, fourth, sixth and seventh holes, getting his bid for a 15th Major championship, and first for four years, off to a flawless start.
The American wasted birdie opportunities before twice finding treacherous rough at the 15th hole to drop his only shot of the day but he was happy with a putting performance that revived memories of him at his pomp.
"I only hit one putt that was off line," said Woods, 36. "I just needed to hit the putts a little bit harder. These greens are not quick.
"With the amount of rain they've had on these things, they're not very fast. So I've got to make that adjustment."
Scott, 32, started steadily but found real inspiration around the turn with five birdies in eight holes.
Three more in a row from the 14th put him in sight of a Major championship record-tying 63 but he found the rough at the last and dropped a shot.
"The calm conditions today were surprising," he said after equalling 1996 champion Tom Lehman's 64.
"It was very pleasing to start off with a solid round because that's what I haven't done at the other Majors this year."
American Johnson, U.S. Masters champion in 2007, made seven birdies but bogeyed the 17th and had to settle for a 65.
Lawrie, the 1999 Open winner, made three successive birdies from the third and picked up further shots at the 14th and 15th to set himself up for a final flourish when a sumptuous iron into the 18th green left him with a tap-in for birdie.
Colsaerts holed his approach for an eagle two at the second and two birdies on the back nine in late afternoon drizzle left him in a share of second place, one ahead of American Brandt Snedeker who stole through the field late in the day with a bogey-free 66.
Former U.S. Open champion Graeme McDowell completed a solid 67, level with Woods, world number two Rory McIlroy, South African Ernie Els, Americans Bubba Watson and Steve Stricker, Japan's Toshinori Muto and Swede Peter Hanson.
Woods calmly birdied the par-three first, picked up another shot with a snaking 20-foot effort on the fourth and pinpoint approaches at the sixth and seventh set up further birdie chances which he converted with immaculate putts.
Stalking the course with unwavering concentration, he looked completely at ease until an errant drive at the 15th left his ball buried in deep rough.
After hacking out awkwardly into another patch of lush deep grass, Woods then produced a brilliant recovery shot to find the green and he narrowly missed a 20-foot putt for par.
McIlroy, among the late starters, moved quietly to three under par before an errant tee shot at the 15th hit a young spectator on the head and ended out of bounds.
The Northern Irishman hastily signed his glove to give to the stricken youth but the ensuing double-bogey hurt McIlroy who dug in bravely to claw shots back at the 16th and 18th holes.
World number one Luke Donald made his first bogey of the day on the last hole but managed only one birdie to card a level-par 70 and stay in the hunt to become the first Englishman to win The Open on home soil since 1969.
Lee Westwood did his chances of achieving that feat no good, however, the world number three failing to take advantage of birdies at the opening two holes and fading badly to finish with a scrappy 73.
Defending champion Darren Clarke also made a poor start with a bogey-strewn 76, former world number one Martin Kaymer slumped to a 77 and four-times Major champion Phil Mickelson had a double-bogey seven on the way to a 73.