The 14-time major champion, in the eighth month of his comeback from spinal fusion surgery after years of back injuries, shared third on eight-under-par after seven holes at Bellerive Country Club.
Scott, who had posted only one top-10 in his last 13 major starts, then pulled his drive way left at the last and could not salvage a par to ensure that Woods finished alone in second place, with Scott's 67 leaving him three adrift of the champion.
Koepka had started the final round two shots ahead of Scott, who was playing with a heavy heart after fellow Australian pro Jarrod Lyle died of cancer earlier this week.
As for Woods? He felt the love, and he gave it right back to St. Louis.
Tiger Woods had three birdies through seven holes as he tried to get in range.
That put him at 10-under 130, breaking by one the PGA Championship record and tying the 36-hole score for all majors.
It looks like we're all going to have a hell of a lot of fun finding out.
Everton complete triple deadline day signings as Bernard joins Barcelona duo
He has signed a five-year contract and joins Lucas Digne, the left back who also moved from the Nou Camp this summer. Everton will open their 2018-19 Premier League campaign against Wolverhampton Wanderers on Saturday afternoon.
Birdies were not only everywhere, but necessary to stay in contention, and Spieth knew it, particularly at 17 - a par 5 where eagle is in play and birdie is a must. "And that might mean shooting 5 under again to hang around, but I would love to be in the mix coming down the stretch and have the chance to hole some putts to win".
Woods has been pressing to play for the US Ryder Cup team but must win the PGA to automatically qualify for the squad going to France next month to defend the trophy against Europe. Kopeka birdied the first two holes to take the lead.
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.
"Hopefully we can come back soon", Tiger said.
He looks up to the task, based off his last couple of tournaments. He missed the birdie putt, but was unfazed.
Three-time major victor Jordan Spieth, needing only the PGA to complete a career Grand Slam, opened birdie-birdie but took a double bogey at the par-3 sixth then followed that with three consecutive birdies.
Of course, the story of the tournament for Woods has been his play on the front nine over the first three days. "Most majors the weekends get hard, but I think this course you're going to have to keep shooting birdies".