After passing up match points in the third and fourth sets, home favourite Kyrgios came out on top in the longest match of his career by beating Khachanov 6-2 7-6 (7-5) 6-7 (6-8) 6-7 (7-9) 7-6 (10-8) at a raucous Melbourne Arena in four hours, 26 minutes.
Top seed Nadal beat fellow Spaniard Pablo Carreno Busta 6-1 6-2 6-4 in one hour, 38 minutes on Rod Laver Arena, while US Open runner-up Daniil Medvedev overcame Alexei Popyrin 6-4 6-3 6-2 and will face Stan Wawrinka in the last 16.
The Swiss advanced after John Isner retired while 6-4 4-1 down, with Dominic Thiem, Alexander Zverev and Gael Monfils joining him in round round four.
CROCKED KYRGIOS CLINGS ON
Kyrgios started confidently against Khachanov and, despite taking a medical time-out at 5-2 up in the first set, managed to edge a second-set tie-break while continuing to grip at the back of his leg.
However, he was unable to repeat the trick at the second time of asking and became increasingly frustrated as the Russian wrested control of the match away from him.
Kyrgios appeared to be struggling for motivation at points and became increasingly vocal, but he hung in to take the victory to the delight of his fervent supporters.
The Australian has a strained relationship with Nadal, who he described as "salty" after their meeting at last year's Mexican Open and impersonated after being called for a time violation while serving against Gilles Simon in round two.
Asked about going up against the 19-time major champion in his on-court interview, Kyrgios said: "Whatever happened between us he's an amazing player. I'm not even thinking about that right now, I'm thinking about my legs and getting them in an ice bath and getting some food."
NO STOPPING NADAL
Nadal was at his imperious best against Carreno Busta, winning 87 per cent and 78 per cent of points behind his first and second serve respectively, while also smashing 41 winners – twice as many as his opponent.
"It was my best match of the tournament so far, without a doubt. Big difference between today and the previous days. I have been serving well, starting to create damage with the forehand," said Nadal.
"Every day is a different story. But, of course, it's important to make steps forward. It's clear that today I made an important one."
Asked if he likes Kyrgios, Nadal said: "I don't know. I don't know him personally, honestly, to have a clear opinion."
RUBLEV ROLLS ON
Rublev started 2020 with titles at the Qatar Open and Brisbane International and has continued his fine form at Melbourne Park.
The world number 16's 2-6 7-6 (7-3) 6-4 7-6 (7-4) triumph over David Goffin was his 15th match win in succession, four of which came in the Davis Cup Finals.
Rublev had wobbles after taking big leads in both tie-breaks, but Goffin made 27 unforced errors in the last two sets to help the Russian progress to a meeting with Zverev.
Seventh seed Zverev is yet to drop a set at Melbourne Park after overcoming Fernando Verdasco 6-2 6-2 6-4.
"IT'S A DISGRACE"
After beating qualifier Ernests Gulbis 7-6 (7-2) 6-4 6-3 to set up a fourth-round meeting with Dominic Thiem – who overcame 29th seed Taylor Fritz 6-2 6-4 6-7 (5-7) 6-4 – Monfils refuted a report claiming excessively playing video games led to him suffering a hand injury.
"You know, as I say, it's misunderstanding with people. I say this in a funny way, to be honest, but it's a disgrace, to be honest with you, because I'm fine," Monfils told reporters in a news conference.
"I'm in the second week, playing great tennis. I think it's just someone who, I don't know, try to — you know, I played ATP Cup, was fine. I practiced here for one month. I was really fine.
"I never said I was hurt. Somehow, you know, playing PlayStation, this? Come on. And it's easy, because I think I'm a good name to make up good story like that. Now you know."