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.
An exhausted Kang said he chose to just relax and have fun despite a lack of sleep, with Saturday's play suspended for darkness and the third round completed earlier on Sunday.
"It was like a normal round of golf for me, it felt like it", said Every, whose two PGA Tour victories were both at the Arnold Palmer Invitational, in 2014 and 2015.
The start of the third round at Trinity Forest was delayed six hours because of heavy rain overnight and into the morning. "Just getting great momentum", said Kang, who matched the course record with a 61 in the second round Friday.
Twice over the last two days, Every overcame deficits of at least four strokes and went ahead of Kang.
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 U.S. when I was 15 and gave me a lot of opportunity to learn from good pros. "I just needed to keep making birdies".
The course was deluged by almost an inch of rain Sunday morning.
It was also a good week for world number three Brooks Koepka, who had an eagle and four birdies in his bogey-free final round to finish fourth on 20-under.
The weather did little to dampen Nicholas Lindheim's day, however, with the American within a chance of shooting his own 61. At 12 under for the tournament, he was one of seven golfers tied for seventh.
"Yesterday we finished too late and I didn't get enough time to sleep". "You know, it just feels unbelievable. We'll see what happens".
But Every had a six-under 30 on the front nine on Saturday to get to 18 under.
Said Every, who played all four rounds with Kang, "I watched him for 72 holes".
A third-round 68 put Kang at 19 under, three strokes ahead of Every (67) going into the fourth round. With birdies at Nos. "He didn't miss. Hard to beat".
The lead changed hands at the par-3 eighth hole when Avery rolled in a 5-foot birdie.
"The good thing is, positive thing is, we only play nine holes and I made a couple mistakes I shouldn't have", Kang said. "We knew it was going to be long day for everyone".
Spieth is still without a top-20 finish in his 13 starts this season.
Asian stocks drop, yuan slides as trade war escalates
The idea that China would dump its $1.1 trillion of Treasuries to retaliate against USA tariffs is often dismissed as improbable. China industrial production and retail sales are slated for Wednesday, same day as USA retail sales and industrial production.