The Australian showman came through a tense battle with the Russian 16th seed after 4hrs 26 mins in front of passionate home support on Melbourne Arena to win 6-2, 7-6 (7/5), 6-7 (6/8), 6-7 (7/9), 7-6 (10/8).
"This is epic, man, I don't even know what's going on," he added. "He's (Khachanov) an absolute warrior," said Kyrgios.
It sets up a mouthwatering clash with the world number one Nadal, who romped past fellow Spaniard Pablo Carreno Busta 6-1, 6-2, 6-4.
Kyrgios, who was handed a 16-week suspended ban in September after a string of indiscretions, disparagingly labelled Nadal "super salty" last year, and mimicked his serving routine during the second round at Melbourne Park.
But Kyrgios tried to play down the feud with Nadal on Saturday, saying: "He has a winning record against me. Whatever happens between us, he's an amazing player, he's arguably the greatest of all time."
The pair, who will meet at Melbourne Park on Monday, last faced each other on grass at Wimbledon in July when Nadal won in four sets.
In other results, Alexander Zverev of Germany beat Fernando Verdasco of Spain 6-2, 6-2, 6-4 and Russia's 4th seed Daniil Medvedev beat Alexei Popyrin from Australia 6-4, 6-3, 6-2.
Austrian fifth seed Dominic Thiem came through 6-2, 6-4, 6-7 (5/7), 6-4 against Taylor Fritz, and Russia's Rublev beat Goffin 2-6, 7-6 (7/3), 6-4, 7-6 (7/4).
Stanislas Wawrinka, the 2014 champion, progressed to the last 16 when John Isner retired in the second set, and Gael Monfils beat Ernests Gulbis 7-6 (7/2), 6-4, 6-3.
Women's matches on Saturday
In the women's matches, Wimbledon champion Simona Halep, produced an impressive display in her 6-1 6-4 win over Kazakhstan's Yulia Putintseva.
Dutch 9th seed Kiki Bertens reached the fourth round in Australia for the first time in her career with a 6-2 7-6 victory over Zarina Diyas of Kazakhstan.
Czech Karolina Pliskova, who won the Brisbane International in the build-up to the tournament, lost 7-6 (7-4) 7-6 (7-3) to Russian Anastasia Pavlyuchenkova.
"I was about 40 percent today of my game," said Pliskova, who was at a loss to explain her lack of form. "I don't know. I was just down."
2016 champion Angelique Kerber, who was made to fight by Camila Giorgi but managed to win 6-2 6-7 (4-7) 6-3.
Switzerland's Belinda Bencic, the sixth seed, won only one game against Estonia's Anett Kontaveit, the world number 31, as she bombed 6-0, 6-1.
And Spain's Garbine Muguruza, a two-time Major-winner, thrashed fifth seed Elina Svitolina from Ukraine, 6-1, 6-2 as six of the top 10 women's seeds exited in round three.
Croatian 19th seed Donna Vekic also crumbled to lower-ranked opposition when she lost 7-5, 6-3 to Poland's Iga Swiatek, who is returning from a stress fracture in her foot.
Belgium's 16th seed Elise Mertens beat Catherine Bellis from the USA 6-1, 6-7 (5/7), 6-0.
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