After completing his storm-delayed second round earlier in the day, Woods thrilled the galleries at the Bellerive Country Club by ripping through the first nine holes in just 31 shots to move within two shots of the lead.
However, his challenge stuttered on the back nine with a run of frustrating pars, wasting an eagle chance by three-putting the 17th, to finish on eight-under 202.
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.
“It’s good but it’s also disappointing at the same time,” admitted Woods after his third round.
“I wish my score was closer to the lead, but not many guys are up there.”
US Open champion Koepka looked set to carve out a big advantage in his bid for a third major in two years, but came unstuck with successive bogeys on the back nine before recovering with a birdie on the 17th to close with a 66 of his own and 12-under 198.
His nearest challenger is former Masters winner Adam Scott of Australia, who carded a fine five-under 65 for 10-under 200, two shots back.
“There’s a big chance tomorrow, but there are probably 10 guys thinking the same thing,” he said.
Spaniard Jon Rahm, halfway leader Gary Woodland and Rickie Fowler, hunting his long-awaited first major, are a further stroke behind.
Woods is bracketed on eight-under with defending champion Justin Thomas, Australian former world number one Jason Day, Ireland’s Shane Lowery and South African Charl Schwartzel.
British Open champion Francesco Molinari is in a three-strong group on seven-under, the Italian continuing the fine form he showed at Carnoustie last month.
But the expected challenge from Jordan Spieth petered out after a triple bogey on the 12th to finish on four-under overall, while current world number one Dustin Johnson also went backwards after a two-over 72 to fall seven shots off the pace.
“I’ve worked my way into a chance to win this tournament just to throw it away on a bad decision,” said Spieth.
Shot of the day came from England’s Matt Wallace, who aced the par-three 16th on his way to a 68 to close on the same mark as Johnson.
Sunday’s play will decide the winner of the fourth and final major of the golfing season with the PGA Championship itself moving to a May date for the 2019 season.