Woods shot a 66 on Saturday, and briefly saw his name sit atop the leaderboard following six birdies in his first 14 holes.
The leader in the Carnoustie clubhouse after the first wave of tee times in the second round of the British Open Friday is named Johnson.
On a day when Woods electrified the gallery with a 66 that pulled the 14-major victor within four strokes of the lead, Spieth, with three majors already, showed again that he is the man to beat of the current generation.
Birdies at the fourth and sixth hoisted Woods to seven-under and left him just a shot behind Spieth and Schauffele in third place after they both came to grief with bogeys on the fifth.
Woods also birdied No. 14 after playing a pitch shot from the back of the green that is shared with No. 4 to move back to 5 under, two off Molinari.
"I felt like I had something to prove to other people with last year's Open and to myself", Spieth said.
Francesco Molinari's win at the British Open has broken the American stranglehold on golf's major championships, with a US golfer having won the previous five. "I need to go get it", Woods said.
Even as Tiger Woods was piecing together a level round, and late starters like Kisner and Jordan Spieth were still on the course, the cut figured to be about 3-over, plus or minus one shot.
One fan said: "That tape on the neck made me a bit nervous. hoping he doesn't blow up on the back 9". "I think the first couple days were hard work and frustrating".
Ideal conditions in the third round led to a number of low scores, including a 64 by Justin Rose and 65s by Spieth and Francesco Molinari.
Golf Channel announcer Nick Faldo added: "I couldn't say which way, if [Tiger's] neck needs relaxing or stimulating, but that's the first time I have ever seen Tiger with KT tape very visible".
But it was to be Molinari's day in the limelight as the red-hot Italian avoided the card-wrecking mistakes that beset his fellow contenders, grinding out par after par and holing a series of clutch putts to keep himself firmly in the hunt.
"I can certainly draw on the all major championship experiences, good and bad that I've had".
Woods was four shots behind. "I was very, very fortunate that it got far enough down there where I had a full wedge into the green".
"As long as 18 is downwind, I don't really care", he said. I can't lie about it, if I could pick one tournament in my life to win, it would be the Open.
"I said to my caddie walking down the fairway, 'Let's have a finish for the crowds at least,"' Lombard said.
It was his best score in a weekend round of a major since the 2010 U.S. Open at Pebble Beach.