Former world No. 1 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 further birdie opportunities before twice finding treacherous rough at the 15th hole and dropping his only shot of the day.
Scott made a steady start to his round but he found 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 told reporters.
"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. The goal today was to play it like a Sunday - and there was no tomorrow."
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 him up for a final flourish when a sumptuous iron into the 18th green left him with a tap-in for birdie.
Former U.S. Open champion Graeme McDowell, inches away from a hole-in-one at the ninth, completed a solid 67, level with Woods, South African Ernie Els, Open champion in 2002, and U.S. Masters winner Bubba Watson.
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.
World No. 3 Lee Westwood birdied the opening two holes but the Englishman, wearing the red cross of St George on his gleaming white shoes, then double-bogeyed the third and he faded badly to finish with a scrappy 73.
Defending champion Darren Clarke also made a poor start with a bogey-strewn 76. World No. 1 Luke Donald is among the late starters alongside four-times major champion Phil Mickelson and Australian Geoff Ogilvy. (Editing by Tony Jimenez)