By Jonathan Jurejko
BBC Sport at Melbourne Park
Nadal, 33, had not dropped a set this tournament before 23rd seed Kyrgios provided his biggest test so far.
Kyrgios, who wore a Kobe Bryant basketball shirt in the warm-up, battled hard but Nadal came through 6-3 3-6 7-6 (8-6) 7-6 (7-4).
Nadal will face Austrian fifth seed Dominic Thiem in the last eight.
Nadal and Kyrgios have not always seen eye to eye with the rivalry blowing up last year after a tempestuous meeting in Acapulco.
Afterwards the 19-time Grand Slam champion accused Kyrgios of lacking respect before the Australian responded by calling him “super-salty”.
Yet the respect for each other’s ability on the court, despite their wildly contrasting styles, has rarely been in doubt and was typified by their firm handshake at the net following a riveting contest on Rod Laver Arena.
“When Nick is playing like he did today with this positive attitude he brings a lot of things to our sport,” said Nadal, who won his only Australian Open title in 2009.
“I encourage him to keep working like this because he is one of the biggest talents of our tour.”