World number one Nadal defeated Mikhail Kukushkin of Kazakhstan 6-4, 6-3, 6-4 and will next face Australian teenager Alex de Minaur.
Marin Cilic suffered a shock Wimbledon second round exit as last year's runner-up blew a two-set lead in his 3-6, 1-6, 6-4, 7-6 (7/3), 7-5 loss to Argentina's Guido Pella on Thursday.
Marin Cilic was crowned Queen's Club champion for the second time as the Croatian stunned Novak Djokovic with a dramatic fightback.
Novak Djokovic reached his first title match for a year with a straight sets win over Jeremy Chardy at Queen's Club.
Highly-rated Canadian teenager Denis Shapovalov was knocked out by Germany's Maximilian Marterer 5-7, 7-6 (7/4), 7-5, 6-4.
The 2014 US Open champion Marin Cilic needed just an hour and five minutes to clinch his third win over the 11th-ranked Spaniard.
Juan Martin del Potro sent his devoted fans home happy Tuesday by powering into the ATP Miami Open quarter-finals with a 6-4, 6-2 triumph over Serbian Filip Krajinovic.
Marin Cilic qualified for the quarterfinals in Miami with his 7-5 and 7-6 (4) victory over Canada's Vasek Pospisil.
Juan Martin del Potro and Marin Cilic are the biggest names left in a field lacking star power.
World No. 8 Del Potro extended his win streak over the Spaniard to five straight matches, winning 6-4, 7-6 (7/3) at the ATP Tour hardcourt tournament here Tuesday.
Roger Federer remained at the number one spot with 10,060 points.
Marin Cilic will play the winner of the first-round match between France's Gael Monfils and Argentina's Horacio Zeballos in the second round.
Roger Federer had previously won the Australian Open in 2004, 2006, 2007, 2010 and 2017.
Cilic will face either defending champion Roger Federer or unseeded South Korean Chung Hyeon, who play on Friday.
World number one Rafael Nadal faces three weeks out after being diagnosed with a torn inner hip muscle during his Australian Open quarter-final defeat
Cilic won 3-6, 6-3, 6-7 (5/7), 6-2, 2-0 in 3hr 47min on Rod Laver Arena and will face Britain's Kyle Edmund for a place in the final on Thursday.
The 33-year-old Frenchman did not disappoint his home crowd as he beat the Croatian in three sets 1-6, 6-3, 6-2 to reach the final of the tournament.
Top seed Marin Cilic from Croatia outplayed Pierre-Hugues Herbert while Kevin Anderson survived a mighty scare from Mikhail Kukushkin.