Smith birdied his first three holes and took three more shots before putting a long squiggly eagle on the 14th green en route to a clear 64 that took him to 13 under par.
US first-round leader Cameron Young was the biggest challenger at 11-under, one shot ahead of Norwegian Viktor Hovland and Irishman Rory McIlroy, who mapped a rock-solid 68 to stay solid in the fight for his fifth major title .
Dustin Johnson was one shot back at nine under, while world number one and compatriot Scottie Scheffler were still one stroke behind Briton Tyrrell Hatton.
Woods needed a huge improvement from his opening 78 to make the cut on the scene of his Open wins in 2000 and 2005.
Continuing his comeback from serious injuries sustained in a car accident last year, the 15-time major champion came alive with a birdie in the third before two bogeys quickly halted his momentum.
He played steadily on the back nine until finding a bunker on the 16th and the ensuing double bogey brought the 46-year-old to nine-over-par, putting an end to any nebulous hopes he had of being back by the weekend. to turn.
Woods received a standing ovation from the stands as he walked up the 18th fairway before missing a short birdie putt to complete a round of 75, leaving him in 148th place in the 156-man field.
“It’s very emotional for me,” Woods told reporters. “I’ve been coming here since 1995. I think the next one will come in 2030 and I don’t know if I’ll be physically able to play by then.
“My two-day game is that I made my share of mistakes,” he added. “Struggled again today to get the feel of the greens.”
GOOD START
Smith, the 28-year-old world number six, this year won the Players’ Championship, the unofficial fifth major, and finished third at the Masters, but his best performance in four previous British Opens was 20th in 2019.
“Obviously I’ve had a really hot start,” he told reporters.
“And it’s very easy to just keep going, be on the front foot and maybe try some shots that are a little too aggressive. But just stayed patient and hit some really nice putts.”
American Young, the world No. 32, bogeyed on the second before taking two birdies to reinvigorate his challenge and birdies on the ninth, 14th and 18th holes helped him to a neat 69.
McIlroy got off to a solid start on his second lap and three straight birdies after the corner narrowed the gap on the leader. He fell back with a bogey on the 15th before a birdie on the infamous Road Hole, the 17th, helping him finish 10-under.
“It was pretty good,” McIlroy said. “It was a very different course to the course we played yesterday with the wind switch.”
Norwegian Hovland rose steadily on the standings, firing his approach shot on the 15th par-four for an eagle, before a birdie brought him to the final level with his Ryder Cup teammate McIlroy.
COMPOSITE JOHNSON
Two-time great champion Johnson shot a controlled 67, recovering from a bogey on the first to ride a hot putter and make six birdies.
“I thought I was playing really well, just riding well, hitting a lot of high quality irons and birdie watching myself a lot,” Johnson told reporters.
“If you look at the history of the tournament, it’s obviously a trophy I’d like to have in the office when I’m done with my golf career.”
Scheffler also bogeyed on the first, but soon regained his usual smooth rhythm to call out four birdies around the turn before signing for a second straight 68.
Hatton battled it out with a bogey-free 66 to finish one ahead of Americans Talor Gooch, Patrick Cantlay and Sahith Theegala plus Australian Adam Scott, the former Masters champion who tore up the Old Course with a sparkling 65.
South Korean Si Woo Kim, Australian Min Woo Lee, Thai Sadom Kaewkanjana and Briton Matthew Fitzpatrick, who won the US Open last month, were among those under six.
A string of former Open winners missed the cut along with Woods, including defending champion Collin Morikawa, Americans Phil Mickelson, Stewart Cink, David Duval and Mark Calcavecchia, Sweden’s Henrik Stenson, Northern Ireland’s Darren Clarke and Scotland’s Paul Lawrie.