The French Open champion belted 26 aces and hammered 62 winners on another sweltering hot day at Flushing Meadows but the 19-year-old Chung held his own against the Swiss fifth seed, ripping 28 winners.
Wawrinka, in his all or nothing, aggressive style, also registered 66 unforced errors against the bespectacled Korean, a 2013 Wimbledon junior finalist who is now ranked 69th.
"It was a really tough match, more than three hours," said the 30-year-old Wawrinka. "He's a great fighter and not easy to play."
Wawrinka, who also has the 2014 Australian Open crown on his resume, next faces Ruben Bemelmans of Belgium, who advanced when American Jack Sock retired due to cramping while leading 6-4 6-4 3-6 1-2.
It was the first meeting for Wawrinka against Chung, who has progressed up the ATP rankings through success in Futures and Challenger tournaments.
The Suwon native has compiled a 49-14 record in Futures and 45-21 in Challengers (4-2 in finals).
Wawrinka said he played better than in his opening-round straight sets win over Spaniard Albert Ramos-Vinolas, but needed three tiebreakers to vanquish Chung.
"He's really good player," said the Swiss. "He's moving well. He's a great fighter. He always tries to give you more balls to play or always tries to find solutions."
Asked if he thought Chung had potential to contend for grand slams, Wawrinka said: "He can be really good, for sure. He's young and already playing well. (But) He has a long way to go.
"You have to see how he's going to practice, improve, how he's going to work. That you will see in a few years only."
On another day of sweltering heat at the National Tennis Center, Wawrinka was relieved to win in straight sets, however challenging they were.
"You can see there are many, many players struggling with the conditions," he said. "It was tough today. More humid, more warm.
"I'm happy to get through. It could have been more than three sets. Could have been four, five sets. You never know. He played some great tennis in those tiebreaks and I'm happy to win this match."
(Reporting by Larry Fine; Editing by Steve Keating)
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