Nadal needed an hour and 37 minutes on court to dispatch fellow Spaniard Albert Ramos-Vinolas 7-6 (7-1) 6-2 and will face Kyrgios later on today.
Nadal dug himdelf out of a 0-40 hole early in the second set by reeling off five straight points including two aces, putting himself up 3-2.
The match was pushed from Thursday night into Friday morning due to heavy rain.
Even though Nadal lost, but the Spaniard will still be confirmed as world number one next Monday when the ATP's rankings points are updated, marking the fourth time in his career he's reached the milestone, and the first time he's reached the summit since 2008.
Roger Federer's withdrawal from the tournament means Nadal will return to world number one for the first time since July 2014, taking over from Andy Murray who is also injured.
"Obviously I think it's easier to say that for me because you see me tank ridiculous amounts of matches against, like, on back courts and stuff like that, and you never see me doing anything silly obviously in a match like this", Kyrgios said.
Later men's quarter-finals pit Japan's Yuichi Sugita against 11th-ranked Bulgarian Grigor Dimitrov, Austria's eighth-ranked Dominic Thiem against Spaniard David Ferrer and John Isner against U.S. compatriot Jared Donaldson.
Pliskova saved two break points and held in the final game of the first set but Giorgi hit a backhand victor on the only break point of the second set to force a third, which the Czech dominated to advance.
However, the Aussie youngster set an upset over the 31-year-old ergo scripting his 100th victory at ATP Tour. I played some good points. And then I played a awful game. "It's those matches for me". Wimbledon champion Muguruza beat Russian eighth seed Svetlana Kuznetsova 6-2 5-7 7-5 and also made a special tribute to victims of the attacks in her country's Catalonia region. But I don't know why I don't find my rhythm here. "But I'm going to keep trying next year". "So no pressure about being world number one".