Edmund falters briefly before progressing but Konta is forced to call time on match
The British No 2 took the first set in a tie-break and withstood a second-set wobble to eventually see off the South Korean.
Edmund will face defending champion Grigor Dimitrov for a place in the last four after the No 1 seed came through a tough test against John Millman.
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Hide AdThe Australian wildcard won the opening set before Dimitrov took the next on a tie-break. The Bulgarian world No 3, who saved two match points in the second-set tie-break, then clinched the decider to win 4-6 7-6 (10/8) 6-3.
Elsewhere, American Ryan Harrison came through his second-round match against Yannick Hanfmann.
He dropped the first set against the German, but fought back to win 6-7 (5/7) 6-4 6-2.
He will face Denis Istomin, who saw off American Jared Donaldson 7-6 (7/5) 6-2.
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Hide AdBritish No 1 Johanna Konta is hoping for a “quick turnaround” after being forced to retire from the Brisbane International.
She had looked in fine form during the quarter-final against Elina Svitolina, but called time on her participation with the Ukrainian leading 1-6 7-6 (8/6) 3-2. Konta appeared to be suffering with a right hip problem and, after seeking treatment two games earlier, quit with what she hopes is just a “low-grade strain”.
The 26-year-old will now turn her attention to getting fit for the Australian Open, which starts on January 15, but will have to wait for an official prognosis.
Konta had not reached a quarter-final of an event since August, but looked to have returned to form after dominating the first set in which she saved all four break points against her serve.
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Hide AdSvitolina twice served her way out of trouble in the second set to force a tie-break, which she eventually won.
A tight third set was interrupted as Konta called for the trainer before she pulled out of the match after two hours and 12 minutes on court.