Down 3-1 in the tiebreak of a fiercely-contested first set, Edmund scored six points in a row to edge ahead in the contest.
Chung, who trailed his rival in the second set, fought back with a decisive break after Edmund blew the chance to go 4-1 up. The 21-year-old pushed on and broke Edmund again in the 12th game to level the match at one set apiece.
Edmund switched gears in the decider, breaking Chung for a 4-3 lead before clinching the match with three aces.
"He got a lot of momentum at the end of the second and broke me twice pretty quickly," said Edmund. "So it was a good third set. I made an effort to really try and step up the intensity and energy and put some pressure on him.
"It was good because the way I was serving in that third set, I knew if I got a break I was going to have a very good chance at holding out."
Uzbekistan's Denis Istomin defeated Jared Donaldson 7-6(5) 6-2 to reach his first ATP Tour quarterfinal since a title-winning effort at the Chengdu Open in September.
Donaldson's American compatriot Ryan Harrison fired 11 aces in his 6-7(5) 6-4 6-2 win over Yannick Hanfmann, the lucky loser who replaced the injured Andy Murray in the draw.
(Reporting by Shrivathsa Sridhar in Bengaluru Editing by Jeremy Gaunt.)
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