The 14-times major champion began the day four shots adrift of fellow Americans Jordan Spieth, Xander Schauffele and Kevin Kisner but with a warm westerly breeze playing havoc, the 42-year-old mounted a thrilling early charge.
Meanwhile, Fleetwood is the most likely to become the first English victor of the Claret Jug since Nick Faldo in 1992.
Woods stirred this remote Scottish town on a balmy afternoon, his name going to the top of the competitors in the 147th Open when he two-putted the 14th green for a birdie.
Meanwhile, England's Eddie Pepperell is safely in the clubhouse on five-under after a four-under-par round of 67.
While the weather on Sunday morning in Carnoustie was warm and sunny, the wind that can be so unsafe here has finally picked up and conditions are set to be tougher over the afternoon. He was tied there with third round co-leader Kevin Kisner.
Tiger Woods of the United States celebrates a birdie on the 9th hole during the third round of the British Open Golf Championship in Carnoustie, Scotland, Saturday July 21, 2018. He managed to punch down the fairway then hit a 60-degree wedge from 83 yards to a few feet to save par.
Woods played with a bit more urgency Saturday, hitting five drivers off the tee after hitting just five through the first two rounds.
Tommy Fleetwood also started his round at 5-under, like Woods, and birdied the first to a massive ovation.
But they'll all be chasing defending champion Jordan Spieth, who had the last laugh after a series of stunning scoring runs from golf's A-list.
And the last two winners came from well back, Paul Lawrie starting the final round 10 shots behind in 1999 and Padraig Harrington being six off the lead in 2007, with both men eventually winning in a play-off.
Tiger Woods walks off the fourth green during the third round of the 2018 Wells Fargo Championship at Quail Hollow Club on May 5, 2018 in Charlotte, North Carolina.
The yellow grass of a dry summer in Scotland comes to life Sunday because of a red shirt at the British Open.
Spieth has his work cut out though, with the leaderboard so closely packed that at least 20 players started Sunday believing they could win the Claret Jug.
Woods, who is playing in his 12th tournament this year after missing most of the past two seasons due to four back surgeries, opened the tournament with consecutive scores of par-71. According to the USA Today, Woods won his last major in 2008 while playing through numerous injuries including a stress fractures in his tibia and torn ligaments in his knee. It's the closest he's been to the lead in a major since five years ago at Muirfield.
Woods was four shots behind.