Through it all, Koepka was calm, making shot after shot when it was needed the most. But Koepka held off all challengers at Bellerive Country Club.
The American finished with a par to post -16 for the tournament, while Scott ended up with a bogey to drop into third place on -13, with Woods securing second on -14.
Tiger Woods may have managed a par for the first time this week, but the 10th hole played as the hardest with 26 bogeys, three double bogeys and a triple bogey from halfway leader Gary Woodland adding up to a scoring average of 4.300. David Toms had the record with his 265 in 2001.
The 25-year-old is without a win since victory in the Open at Royal Birkdale last year and 12 months on there would be no repeat at Carnoustie, where he went into the final round with a share of the lead but stumbled to a closing, birdie-free 76.
"Looking where I was, sitting on my couch watching the Masters, and to think I would do this, I would have laughed at you and told you there was no way, no chance, and to do it is really incredible.
"I need to assess where I'm at".
He never imagined what it would take for him just to get to the weekend. Every shot is so important, and when you're out here grinding away, firing at the flags, firing at the middle of the greens, you just have to be very patient.
Though he has not been in good form recently, the 48-year-old will probably still be picked by Furyk, who has a few more weeks to ponder his choices.
The chances of a controversial victor with Adam Scott just two off the lead.
Tiger Woods walks the 18th fairway during the final round of the 2018 Wells Fargo Championship at Quail Hollow Club on May 6, 2018 in Charlotte, North Carolina. The birdie at 8 moved him to within two shots of the lead at the time.
He bogeyed the last to shoot three-under-par 67 and finish three strokes behind Koepka's 16-under 264.
Coming in second at -14, two shots behind Brooks Koepka, also left Woods with one hell of a consolation prize.
"He was unflappable. "[My coach] Claude didn't say one word to me this week, I was hitting it that good", Koepka said Sunday.
I don't think I'd ever played in front of that many people.
But for all the buzz Woods generated on Saturday, the reality is that capturing a 15th major from four shots back of the front-running Brooks Koepka will require something special.
But then the golfing gods gifted this marathon Saturday out of left field and left us all with a Sunday at colorless and faceless Bellerive that could be the most memorable golf event of the year, and of many years to come.
"I played pretty well". "I was always trailing", Woods said. I struggled to hit it as the greens were getting slower.
"So from there, it's been phenomenal what I've been able to do in my mind", he said.
Koepka with a chance for birdie on the 16th, to send him two shots clear of Scott, three ahead of Tiger. You're never going to win a major. He had a two-shot lead heading into the back nine.
As the putts kept dropping, the roars got louder. "It was enjoyable to play in front of fans who love golf and who are cheering for us".
"Could've been a little closer", Woods said.
"I feel like I'm extremely confident".
"It was a course I couldn't quite sit still and be okay with it. So that's a bummer". He shot 68 and tied for sixth. World No. 1 Dustin Johnson carded a 69 in the final round, leaving him in a tie for 27th. It was his seventh runner-up finish and first since the 2009 PGA Championship.