"Today we could be a little more offensive, which was nice".
Indeed, Spieth is right in the mix himself, along with fellow Americans Kevin Kisner and Zach Johnson, who shared the overnight lead on six under par. Rose is No. 3 and almost joined them until he made an 18-foot birdie putt on the last hole to make the cut on the number.
Kisner made a tough par save on the 17th and saved par again from behind the 18th green for a 68.
They were at 9-under 204.
On a day that was flawless for low scoring, Woods let rip with his driver and tucked into the birdie buffet with his red-hot putter, briefly grabbing a share of the lead - something the American had not done at a major since the 2013 Masters.
Woods claimed not to have noticed he had spent about 20 minutes atop the board. But it was there.
England's Justin Rose is a shot behind Woods after earlier equalling the lowest score in an Open at Carnoustie with a brilliant 64.
Spieth, the defending British Open champion and a three-time major victor, didn't trail the leaders for much longer, birdieing the fourth hole to move into a tie for first at 6 under.
He said: "I probably just didn't give myself enough time to regroup after a photographer was in a world of his own". When I hit great drives, I couldn't quite get the irons close.
Kevin Chappell, who spent most of his round watching Spieth put on a show, birdied the 18th for a 67 and was two shots behind. After parring the first, he drifted wide on the second, and ended up in a situation where he needed to fire a shot right over the gallery. They all dropped shots on a day when there was no time to be going backward. He shot 72 and was in the group four behind.
Perhaps none of the leading men will go full Jean Van de Velde, the star-crossed Frenchman who could still be standing in that burn if not for a rising tide back at the 1999 championship, but if the 499 yards of dusty turf is an uninvited guest, it's a guest nonetheless. The biggest advantage he sees is to "expect the unexpected", and the most windy conditions of the week might make it every bit of that. Spieth, who spent Saturday morning watching The Open on television, asked caddie Michael Greller on the practice range, "Do we like driver?" "I really, really did". I felt I really had control of the ball'. "Even when you think you've got it, it will jump up and bite you". At the long par-three 16th his tee-shot just failed to stay on the green and he three-putted.
The last time Woods led or held a share of the lead at the end of a round at a major was in 2012, at the US PGA Championship.
Kisner had the 54-hole lead at the PGA Championship a year ago.
"It's insane. Absolutely crazy to think so many people can follow a person", he said.
Tiger Woods is out on the course, making a par on the first hole.