Disgraced world number one Woods, welcomed back from self-imposed exile by cheering crowds at Augusta National, soared into contention with two eagles on the way to a four-under-par 68 to lie two strokes off the pace.
"It's unbelievable," Woods told reporters after breaking 70 for the first time in the opening round at Augusta in his 16th Masters appearance. "I would think that a couple under par would have been probably a pretty good start.
"Everybody seemed to go low today and there are 30-plus guys under par. So the golf course, even though it was windy, it could be had today. If I putted well today, it could have been a really special round."
Asked about his rousing reception by the fans, Woods replied: "It was unbelievable all day. I haven't heard them cheer this loud in all my years here."
While the 14-times major winner drew the day's biggest gallery, 1992 champion Couples fired a sizzling 66 in breezy conditions to take command of the year's opening major.
Maintaining the loose-limbed form that has swept him to victories in his last three starts on the seniors tour, the 50-year-old American birdied four of the last seven holes to break clear of the field.
On a gripping day featuring sudden weather changes and electric displays by several former champions, 60-year-old Tom Watson defied his age to share second place on 67 with fellow American Phil Mickelson, Britain's Lee Westwood and South Koreans Yang Yong-eun and KJ Choi.
American Anthony Kim birdied the last three holes for a 68, finishing level with compatriots Woods, Nick Watney and Ricky Barnes, and British world number seven Ian Poulter.
The main focus amid the Georgian pines and heavily contoured layout was on four-times champion Woods, who looked remarkably composed as he started one of the most eagerly anticipated rounds of all time under intense scrutiny.
Despite not having competed since winning the Australian Masters on November 15 following stunning revelations about his extra-marital affairs, he raced to the turn in three-under following two birdies, a bogey and an eagle at the eighth where he coaxed in a curling eight-footer.
Woods, who tipped his cap to the supportive crowds on every hole and repeatedly said "thank you," bogeyed 10 and 14 but comfortably birdied 13 before rolling in a nine-footer to eagle the par-five 15th.
The 34-year-old, who had never before recorded two eagles in one round at Augusta, raised his putter in celebration before lipping out with a four-foot birdie attempt at the last.
With his ball-striking in impeccable form despite his length break, Woods is well positioned in his bid to become the first player to launch his season with a major victory since fellow American Ben Hogan clinched the 1953 Masters.
Under leaden skies, the galleries were packed at least five deep all the way down the opening hole to watch the world number one start what media experts predict will be the biggest single day television audience for a U.S. golf event.
While the world number one had to be feeling a little anxious, he offered a warm smile as he shook hands with playing partners Choi and American Matt Kuchar on the first tee.
The official starter announced: "On the tee, Tiger Woods," prompting loud cheering by fans crammed on either side of the fairway in front of the imposing clubhouse at Augusta.
There were calls of "Go Tiger" and "We love you Tiger," not a hint of heckling and plenty of hooting and hollering in the loudest applause the world number one has received all week.
While Woods began his opening round in strengthening winds, eight-times major winner Watson was putting the finishing touches to a vintage display.
Watson, who came agonizingly close to the most remarkable major victory of all time before losing the 2009 British Open at Turnberry in a playoff, rolled in a five-footer to birdie the last before throwing his arm skywards.
"I did what I had to do today and took advantage of some of the holes that you could take advantage of," Watson said.
Watson's trip down Memory Lane was eclipsed late in the afternoon by Couples, who has long regarded the Masters as the biggest event in golf.
"The Champions Tour has been a lot of fun, but this is where I really want to play well," Couples said. "Today was as good a round as I've ever played here."
© 2013 Thomson/Reuters. All rights reserved.