The mercurial Australian unleashed a profanity-laced tirade at the chair umpire, Fergus Murphy.
Kyrgios won the opening set despite foot and ankle issues, but after a rant he threw his towel over a camera, forcing the broadcasters to switch to an overhead view.
After Khachanov won the second set, Kyrgios shouted: "The worst f***ing ref ever, the worst ref in the game.
"The worst, hands down. Every time I play he's doing some stupid s***, every time."
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At the end of the match Kyrgios unleashed on Murphy again, saying: "You're a f***ing tool bro."
Khachanov now moves on to the third round and will take on Frenchman Lucas Pouille, who beat Canadian Denis Shapovalov 6-4, 6-4.
Kyrgios is likely to face disciplinary action for his outbursts.
Earlier, Alexander Zverev hit 20 double faults in a bruising loss to 19-year-old qualifier Miomir Kecmanovic.
The German has had a poor season and this was another low point, Kecmanovic benefiting from his opponent's wayward serving in a 6-7 (4), 6-2, 6-4 victory, his first over a top-10 player.
The bottom half of the draw is wide open, with fourth seed Kei Nishikori another big name to fall.
Nishikori lost an all-Japanese clash with Yoshihito Nishioka, who defeated a man he calls his hero 7-6 (2), 6-4 for the biggest victory of his career.
Nick Kyrgios could face disciplinary action for his latest outburst
(Reuters)
Third seed Stefanos Tsitsipas is in the top half but was not immune to the upsets, recovering from a set and a break down against German Jan-Lennard Struff only to lose 6-4, 6-7 (5), 7-6 (6).
Seventh seed Daniil Medvedev continued his good form by beating Benoit Paire 7-6 (2), 6-1 but Stan Wawrinka was ousted 6-4, 6-4 by qualifier Andrey Rublev.
PA
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