Taking advantage of ideal wind-less scoring conditions at the Old Course, Daly was level with fellow American Sean O'Hair, Northern Ireland prodigy Rory McIlroy and also-finished Scotsman Andrew Coltart for the early lead.
Daly, who won the 1995 Open at St. Andrews, set himself up as a people's choice to hoist the Claret Jug just as Tom Watson did last year and Greg Norman had in 2008, although both of them failed in the end.
The long-driving 44-year-old American opened and closed the front nine with back-to-back birdies and birdied the par-5 fifth as well to make the turn in 31 over the legendary links in the event's 150th anniversary edition.
Daly sank a 10-footer for birdie at the 10th and birdied the par-3 11th before lipping out a five-footer at the 12th to end his birdie streak and begin a set of birdie near-misses that denied him an even better round.
At the Old Course's famous 17th hole, the Road Hole whose tee was moved back 40 yards for the Open, Daly found a path between the green and the road with his approach and missed a 15-foot putt to take his first bogey of the day.
Another lip-out for birdie at 18 left him only level with his best prior round of the year from the first round at Colonial in May.
Turning back the clock for a third major title would be a stunning feat by Daly, whose struggles with weight, women and alcohol have made him a fan favourite but taken a toll on his game.
Daly, whose other major title came at the 1991 PGA Championship, has not won since 2004 and his status reduced to taking US PGA Tour sponsor exemptions. He is ranked 455th and has only once cracked the top 55 on the US tour this year.
Daly, who beat Italy's Costantino Rocca in a four-hole playoff 15 years ago at St. Andrews, wore a pink shirt and light blue vest with purple and green pants, a shock-style fashion that has been a recent trademark.
If anyone was going to have a somewhat redemptive triumph this week, the money literally was on Woods, a betting favourite despite being winless since a five-month hiatus caused by a sex scandal that shattered his iconic image.
Woods, paired with England's Justin Rose and Colombian Camilo Villegas, was greeted with polite applause, similar to his receptions at the year's earlier majors, where he shared fourth each time.
World number one Woods, a 14-time major champion chasing the record 18 major titles won by Jack Nicklaus, seeks a fourth British Open title after having won at St. Andrews in 2000 and 2005 and at Royal Liverpool in 2006.
Woods sank a 25-foot birdie putt at nine to reach three-under, where he stood after 11 holes.
McIlroy, 21, eagled the par-4 ninth and began the inward nine with three birdies in a row and was level for the lead with five holes to go. He missed the cut at this year's Masters and US Open but seems poised for a solid run.
O'Hair opened with three birdies and closed the front nine with birdies on three of the last four holes to share the lead with six to play.
Germany's Marcel Siem was in the clubhouse of 67.
Siem, whose only title in eight European Tour years came at the 2004 Dunhill Championship in South Africa, sizzled with five birdies in his first 10 holes on his 30th birthday but the world number 251 was only level par from there.
The winner over some of the most hallowed ground in sport will take home 850,000 pounds (one million Euros) and have his name engraved on the Auld Claret Jug trophy alongside some of golf's greatest legends.
Irishman Padraig Harrington, bidding for his third British Open title in four years, put himself in the Swilcan Burn and opened with a double bogey.
World number two Phil Mickelson, who won his third Masters and fourth major title in April, goes off later. The US left-hander can overtake Woods and become world number one for the first time with a victory.