Koepka will look to keep the momentum when he defends his PGA Championship title at Bethpage Black in Farmingdale, New York next week.
Kang, who said he unexpectedly felt calm throughout the round, had just three-putted for his first bogey in 38 holes. So while Every was the overnight leader, Kang held a three-shot advantage going into the final round.
Kang matched the course record with a 61 in the second round to take a four-stroke lead over Every into Saturday after the two also played together the first two rounds.
The hot play continued across the back nine when a 23ft birdie on the par-four 15th - the second of three straight birdies - gave Kang the outright lead. "He didn't miss. Hard to beat", Every said.
Spieth is still without a top-20 finish in his 13 starts this season.
"Today's round wasn't great, but overall a pretty solid week, especially on the greens", Spieth told CBS after his round. That put him 3 strokes ahead of playing partner Matt Every, who had four consecutive bogeys on the back nine in an up-and-down round of 67.
Every had edged ahead by one through nine holes when darkness halted play on Saturday.
Every, who started that 431-yard par 4 tied for the lead, was in the middle of the fairway before what he called "just a bad shot" on his approach that set up a three-putt for bogey.
"We knew it was going to be a long day so just need to be patient", said Kang, 31.
Asked about a telephone conversation with his father in South Korea, where he was heard to say "I did it", he added: My dad gave me a lot of support when I was young and sent me over to the United States when I was 15 and gave me a lot of opportunity to learn from good pros.
Spain's Carlos Ortiz made a big move in the final round, with an eagle on the first hole and going 5 under through seven holes. But he was only one under through 10 holes when play stopped.
But on Sunday at Trinity Forest Golf Club, Kang stayed strong despite a chaotic final day.
Sung Kang waves to fans as he arrives at the 18th green during round 4 of the AT&T Byron Nelson golf tournament on Sunday, May 12, 2019 at Trinity Forest Golf Club in Dallas.