Four-time Grand Slam champion Osaka advanced after Ukraine's Lesia Tsurenko, who beat Ons Jabeur in round two, pulled out with an elbow injury.
Osaka will meet another former world number one Karolina Pliskova for a place in the last four, as the Japanese star continues her comeback from maternity leave.
She is through to her first WTA quarter-final since reaching the Miami Open final almost two years ago.
Pliskova, who won her first title in four years in Romania last week, fought back from the brink of defeat to overcome Czech teenager Linda Noskova 3-6, 7-5, 6-1.
Noskova served for the match up 5-4 in the second set before the wheels came off against two-time Grand Slam finalist Pliskova.
Chinese fifth seed Zheng let slip a strong position in the opening set where she led 5-3, as Fernandez broke twice in a row and reeled off the final four games.
Fernandez then punished a loose service game from Zheng in the sixth game of the second set before closing out an impressive victory.
"I'm extremely pumped up, I'm super happy with my level. There's still a lot of work to do, but so far so good, so we're just going to keep going," said Fernandez.
World number 38 Fernandez, a former US Open finalist, goes on to meet third seed Elena Rybakina for a spot in the semi-finals.
Rybakina withstood a spirited fightback from 16th-seeded American Emma Navarro, winning 6-1, 6-7 (6), 6-4.
The Kazakh had a match point in the second set tie-break but the in-form Navarro, this year's Hobart champion, forced a deciding set.
Rybakina broke for a 3-2 lead and saved two break points in the final game to seal the win.
"It was such a difficult match," said Rybakina. "I had some opportunities in the second set but she played really well and for me physically it was a little bit more difficult. It was tough."
Anastasia Pavlyuchenkova also progressed to the last eight with a 7-5, 6-3 win over Wimbledon champion Marketa Vondrousova.
Katerina Siniakova, who took out Coco Gauff in the second round, could not build on Tuesday's success as the Czech lost in straight sets to Danielle Collins.
Top seed Iga Swiatek also progressed later on Wednesday, easing past Ekaterina Alexandrova 6-1, 6-4.
