Having led Karen Khachanov by two sets and a break, Kyrgios eventually won on a deciding fifth-set tie-break after the longest match of his career, while Nadal cruised to victory over Pablo Carreno Busta.
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Nadal revealed he was caught up in the drama of Roger Federer's epic five-set victory over John Millman on Friday night. The top seed stayed up until 1am to see how it played out, saying: "It was impossible to sleep watching the match. It was emotional at the end. I enjoyed it. Anything could happen, but Federer is Federer."
Thomas doesn't pass Muster
Dominic Thiem added former French Open champion Thomas Muster to his coaching team ahead of the Australian Open but the partnership has not even lasted the tournament.
Thiem, whose main coach is former Chilean player Nicolas Massu, told the German-speaking media: "We ended our working relationship. It did not fit. It's easy like that. It is just about work. We get along well personally and we said in the beginning if anything is not all right we will tell each other."
Women: Karolina Pliskova (2), Elina Svitolina (5), Belinda Bencic (6), Donna Vekic (19)
Men: David Goffin (11), Karen Khachanov (16), John Isner (19), Pablo Carreno Busta (27), Taylor Fritz (29)
Who's up next?
Federer will try to made the most of his second chance when he takes on Hungary's Marton Fucsovics in the fourth round on Sunday.
Earlier in the night session, top seed Ashleigh Barty will look to avenge her Wimbledon loss to Alison Riske while Coco Gauff faces fellow American Sofia Kenin as she tries to reach a grand slam quarter-final for the first time.
Defending champion Novak Djokovic faces 14th seed Diego Schwartzman and Serena Williams' conqueror Wang Qiang takes on Ons Jabeur.