Six other major winners, including 14-time major champion Woods, lurked within three strokes after an early birdie as an afternoon shootout began to unfold at Bellerive Country Club.
Those errors brought Woods back into contention.
Woods started the day with five birdies in his first eight holes. He tapped in for par on the final hole to set the PGA Championship scoring record at 264.
"It's a good position to be in and there's a lot of holes to be played", Schwartzel said.
And Woods' last chance to apply real pressure vanished when he pushed his tee the par-5 17th right of the creek running along the right side of the hole.
"I'm exhausted. I am definitely exhausted", Woods said.
Now Koepka has to hold off a wide circle of PGA contenders that's heavy on instant name recognition and, in Woods, that includes arguably the greatest golfer ever trying to write one of the game's greatest tales.
Scott hung around by making big putts, just like he hoped, and was tied for the lead until Koepka's birdies.
Woodland is the only one to make a triple bogey on No. 10 in the event, but the par-4 hole was the hardest on the course Saturday with a 4.3 average, compared to the easiest, the par-5 17th (4.45).
At the 10th, Woods found the right rough and pitched to eight feet then made a tension-packed putt to rescue par, a right fist-pump showing his joy at avoiding a third consecutive bogey on the hole. "All in all, we'll play aggressive and just stick to what we've been doing". But he answered with four solid rounds at Bellerive, capped by a 69 Sunday that left him with a top-20 finish, while closest pursuers Bryson DeChambeau and Phil Mickelson missed the cut. He outlasted good friend Dustin Johnson at Shinnecock Hills to become the first back-to-back US Open champion in 29 years.
He looks up to the task, based off his last couple of tournaments.
"It's been a pleasure to play in front of them, it really has", Woods said. "I didn't do that, I rapped it past the hole and missed the putt".
The four-time PGA Championship victor made five birdies and one bogey through eight holes and finished with 10 consecutive pars.
It left a sense of anti-climax, but Woods is still within reach of his 15th major, a situation few would have predicted at the start of a year which has seen the 42-year-old resurrect his career after debilitating injuries.