The 18th seed raced out to a 4-1 lead before Korda calmed his nerves to force a tiebreak but the Russian hit back to win the first set with a backhand scorcher.
Korda, whose father Petr had triumphed at Melbourne Park in 1998, struggled on his forehand and had his right wrist strapped midway through a tight second set before Khachanov broke to go up 4-3 and the 26-year-old held firm to double his advantage.
Sebastian Korda has been forced to retire due to injury.@karenkhachanov advances 7-6(5) 6-3 3-0 ret.#AusOpen •… https://t.co/bnc0XuujBh
— #AusOpen (@AustralianOpen) 1674539547000
With his injury affecting his rhythm and the match slipping away, 29th seed Korda dropped serve early in the third before throwing in the towel.
Khachanov will meet Stefanos Tsitsipas or Jiri Lehecka in the last four.