Kyrgios brushed aside the challenge of seeded Italian Fabio Fognini to cruise to a 6-2 6-4 victory in just 61 minutes before Kokkinakis needed almost three hours to shake off American Denis Kudla 7-6 (7-5) 4-6 7-6 (7-4).
Fognini was out of sorts in the opening set, struggling with his movement, as Kyrgios took full advantage with service breaks in the opening game and again in the seventh.
Italy's world No.34 rallied in the second set despite appearing uncomfortable between points, but perfectly fit during play.
The Australian, eight years younger at 26, but ranked 102nd, was frustrated at this inconsistency as the match went with serve to 4-4.
Kyrgios then stepped up a level to break serve before he closed out the victory.
"I wanted to keep the momentum going," Kyrgios said.
"As soon as I had break points, I wanted to take them.
"I served well and I am happy to be through. I am just returning well.
"My doubles has helped me a lot on my singles court. The way I am serving and returning is quite a sight to see."
With his victory, Kyrgios has improved to 7-2 on the season, having arrived in Miami on the back of a run to the quarter-finals at Indian Wells.
He next faces world No.11 Jannik Sinner, a 5-7 7-5 7-5 winner over the 19th-ranked Pablo Carreno Busta, for a spot in the quarter-finals.
"It was a really great performance, not only was his tennis of a really high standard I thought mentally he kept his composure really well," British commentator and four-time Wimbledon semi-finalist Tim Henman said of Kyrgios.
"It was mature, professional, calculated. He'll be extremely pleased with his progress so far."
Kokkinakis followed his Australian Open doubles-winning partner into the last 16 with a spirited display against Kudla.
He stood three points from losing the marathon encounter before pulling out an important win that will likely set up a showdown with German second seed Alexander Zverev.
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