That left Woods five strokes off the pace of USA compatriot Koepka, who birdied five of his first nine holes to make the turn on 13-under par, three shots ahead of Americans Rickie Fowler and Gary Woodland with Australian Adam Scott, the 2013 Masters champion, and American Kevin Kisner another stroke adrift.
Even with two bogeys, Woods shot 64 for his lowest final round in a major.
The 42-year-old then got everyone on the edge of their seats at The Open when he briefly sat atop the leaderboard in the final round before finishing tied-sixth. Spieth didn't get under par for the tournament until his seventh hole Friday - the par-3 16th hole - and he managed to do enough right for a 66 to get within seven shots of the lead. With only half of the 156 golfers finished with their second round, a heavy rain storm hit the course and suspended play at the PGA Championship until 7 AM Saturday morning.
The two Utah club professionals in the field each ended up at 5 over par, missing the cut.
Woods, in the eighth month of his comeback from spinal fusion surgery after years of back injuries, began four back.
Just look at this video of Woods arriving at Bellerive for the final round.
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"As much as I can", Brown said of trying to follow Woods throughout his round. His 72-hole score of 264 set the PGA Championship scoring record and matched the major championship record set by Henrink Stenson at Royal Troon in the 2016 British Open.
Resuming his weather-delayed round on the 12th hole Saturday, the Lehi resident made two more birdies and two bogeys to finish. Seemed like everybody was between 2- and 4- or 5-under par. "All in all, we'll play aggressive and just stick to what we've been doing".
Koepka never imagined a year like this. "It was nice to have that in my back pocket, (having) been there in that same situation, and came out on top".
Koepka joked about working out in a public gym this week with Dustin Johnson and not being recognised.
After failing to hit a fairway in regulation on the front nine, the four-time PGA Championship victor needed only 10 putts in an extraordinary display of scrambling.
Fleetwood signed off with a 68 for tied 36th place.
For the record, Woods didn't hit a fairway on the front nine.