The two men shared the first two sets, before Nadal was able to edge an attritional third through a nervy tiebreak.
Nadal controlled long spells of the fourth set but spurned the chance to serve for the match, with Kyrgios doing brilliantly to cling on. The top seed got a second chance in another tiebreaker and eventually found a way past his battling opponent.
The win for Nadal means he will face fifth seed Dominic Thiem in the last eight as he goes in search of his 20th Grand Slam title overall and his first win at this event since 2009.
Before the match, Kyrgios paid tribute to basketball star Kobe Bryant, who was killed in a helicopter crash on Sunday:
When the tennis got started, the home player was given raucous backing by the Melbourne crowd. However, the atmosphere didn't daunt Nadal, who was able to quickly settle into an ominous groove.
The top seed was able to conjure this magical forehand down the line early on and grab a break of serve:
By contrast, Kyrgios was unable make a dent in his opponent's serve and was powerless to prevent the 19-time Grand Slam champion from clinching the first set.
After being comprehensively outplayed early on, it was imperative Kyrgios started the second set well. He had to dig deep to hold serve against a Nadal barrage, and that appeared to fill the Australian with confidence.
Kyrgios then broke to move 3-1 in front and behind his booming serve he was able to do enough to level things up with the second set. Eleanor Crooks of the Press Association noted that Nadal was looking a little unsettled at this stage:
In the third set, both players settled in behind their serve and there was just one break point carved out between the pair; that was for Nadal, although Kyrgios was able to hold.
It meant a tiebreak was needed to separate the players and a tense hush fell over Rod Laver Arena for it. After falling 3-1 behind in the breaker with a loose shot, the Australian took out his frustration on his racquet:
Despite dropping the next point, Kyrgios managed to pull himself back into it. However, Nadal clung on and muscled his way into the lead again. George Bellshaw of Metro offered his thoughts at the end of a draining third set:
At the start of the fourth, it appeared as though that huge effort had taken something out of Kyrgios and in the third game of the set, sloppy play from the Australian allowed Nadal to secure an early break.
Ben Rothenberg of the New York Times was losing faith in Kyrgios' chances of getting back into the match:
While Kyrgios was slowly wilting, Nadal was continuing to play relentless tennis, as he manoeuvred his opponent around the court with trademark intelligence.
Backed by the Australian crowd, Kyrgios continued to cling on in his service games and forced Nadal into serving the contest out. Although that scrapping from the home player looked like it would be in vain, Nadal got some jitters with the ball in hand and allowed the Australian to get back on serve.
However, even with that impetus, Kyrgios couldn't force a fifth. In the tiebreak, Nadal was able to show more match awareness and discipline on the crucial points, ensuring he eventually finished a draining affair in four sets.