Koepka seemed to be cruising to his fourth major title until he ran up four successive bogeys as Bethpage Black reared its difficult head and bit the leader in strong and gusty winds.
The American started the day with a seven-shot lead and was still in command until bogeying the 11th, 12th and 13th holes.
World number one Johnson, playing two groups ahead, made a brilliant birdie at the difficult 15th, rolling in a 10-foot putt that cut the margin to two shots.
With the raucous crowd openly chanting "D.J." in Koepka's face, the seemingly unflappable leader suddenly looked all too human.
Moments after Johnson's birdie at the 15th, Koepka air-mailed the green with his tee shot at the par-three 14th and made another bogey.
A championship that had been largely devoid of drama since Koepka ran away from the field on Friday sprang to life.
But there was another twist when Johnson bogeyed the 16th hole, missing an eight-foot putt as the pendulum swung back in Koepka's favour.
TOUGH CONDITIONS
Earlier, Koepka handled the tough conditions and the pressure of leading with ease.
One bogey and one birdie on the front nine left him as he started the day at 12 under par when he made the turn, with Johnson his only realistic threat.
Koepka's seven-shot lead starting the day did not last long, as he bogeyed the par-four first, where he found deep rough with his drive.
The dropped shot left him only five clear of playing companion Harold Varner III, who birdied the hole.
But Varner quickly dropped out of contention with successive double-bogeys, leaving Johnson as Koepka's only danger.
A victory would be Koepka's fourth major in less than two years, following the 2017 and 2018 U.S. Opens and the 2018 PGA Championship.
Johnson has only one major to his credit, the 2016 U.S. Open.
EARLY SCORES
There were some good early scores on Sunday morning before the wind whipped up, with American Beau Hossler shooting a six-birdie, four-bogey two-under-par 68.
But the course average was some three shots over the par of 70.
Phil Mickelson was among the casualties on Sunday, battling his way to a six-over-par 76.
"It's certainly challenging but there's nothing over the top because the greens are somewhat receptive," he said, referring to the soft putting surfaces.
"I thought it was a great set-up to identify who's playing the best."
Mickelson finished at 12-over 292, but the American was a clear number one with the rowdy fans.
As boisterous as the spectators have been this week, one can only imagine what it will be like when Bethpage hosts the 2024 Ryder Cup, when Mickelson could be captain on the home team.
(Reporting by Andrew Both, editing by Pritha Sarkar and Ken Ferris)
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