An inspired Watson, at the age of 53, eagled the par-four 12th and birdied two of his last three holes to record his lowest score at a US Open since 1987 on "a very special day".
Justin Leonard, the 1997 British Open champion, and fellow American Jay Don Blake were a further stroke back on 66 while Australia's Stephen Leaney and Jim Furyk shared fifth place on 67.
"Who would have thought, who would have expected that I would have shot a round like that today," Watson said after he bounced back from a bogey-five at the 10th, his first hole, with some magical golf.
"I came here to play my best and I'm hitting the ball well enough. It's just a matter now of being able to deal with the pressure well enough at our national open."
Woods, bidding for his third US Open crown in four starts and the ninth major of his career, initially struggled on the difficult greens at Olympia Fields Country Club and was one over par after 14 holes.
But the world number one, who won last year's tournament by three shots at Bethpage Black, resurrected his round by hitting a superb three-iron approach to 20 feet at the 555-yard sixth hole before sinking the eagle putt.
In overcast conditions, he then missed a birdie chance from 12 feet on seven, scrambled a par on eight and bogeyed nine after finding a greenside bunker and failing to get up and down.
That left the 27-year-old fives strokes off the early lead, but importantly still in the tournament.
"Consequently, I fired away from just about every flag.
"In a major championship, you just have to keep plodding along. I made 15 pars today, and unfortunately I had a couple of bogeys.
"I didn't hit the ball off the tee as well as I needed to, but I got it around and kept myself in the tournament."
Woods had teed off with twice winner Ernie Els and U.S. amateur champion Ricky Barnes at the par-four 10th, reeling off eight pars before bogeying the par-four 18th, where he pulled his drive under the lip of a fairway bunker.
British Open champion Els, winner of the 1994 and 1997 US Opens, salvaged pars on his first two holes after twice finding fairway bunkers off the tee, and then moved to one under with a birdie-three at the 414-yard 14th.
The big-hitting South African, who won four titles in his first five starts of the season, solidly parred the remaining holes for an opening 69.
"The greens are really difficult to putt, especially if you get past the hole," said the 33-year-old Els.
"But I'm happy with the way I played. I didn't get much out of it but anything under 70 is good for the first round."
The 33-year-old Quigley, yet to win on the PGA Tour, did not make the halfway cut in his two previous U.S. Open starts but stormed into contention at Olympia Fields in the afternoon sunshine.
He mixed three birdies with a bogey to reach the turn in 34 and, after dropping another shot at the par-four 10th, stormed home with four birdies in the last seven holes to finish the day at five under.
"I never felt nervous today and just played well. I felt good with the putter and managed to keep the momentum going all the way," said Quigley, who missed five successive cuts on this season's PGA Tour.