The fourth-seeded Zverev had battled through a pair of three-set wins in his opening two rounds but was keen to make short work of Kyrgios when he broke serve in the first game and stormed to a 5-1 advantage.
The 22-year-old Kyrgios battled back to win three straight games to make it 5-4 but Zverev held his nerve and serve to close out the opener in 34 minutes.
Kyrgios, who had reached the Miami semi-finals in the previous two years, saw his hopes of at least matching that feat evaporate in the second set after his opponent secured the only break in the fifth game and rode it out for a well-earned win.
Earlier, big-serving American John Isner overcame second-seeded Croatian Marin Cilic 7-6(0) 6-3 to advance to the quarter-finals, where he will face South Korean Chung Hyeon, who dominated Joao Sousa in a routine 6-4 6-3 win.
In a battle of power against power, it was Isner's booming serve that made the difference as the towering 32-year-old fired 12 aces while never giving Cilic a look at a break point on a hot, windy day.
"I was very happy with how I played, especially with how I served," Isner told reporters. "That's certainly the best I've served all year."
DEL POTRO TRIUMPHANT
Isner took firm control of the match in the first set tiebreaker, where he stepped up his return game to shut out the world number three before finally breaking Cilic's powerful serve in the second set and cruising to the win.
Chung needed a little more than an hour to see off the Portuguese Sousa, who could not find an answer to the 19th-seeded South Korean's relentless all-court game.
With the win, Chung advances to his second consecutive quarter-final after also making the last eight in Indian Wells, where the 21-year-old lost to Swiss Roger Federer.
Also advancing was Juan Martin del Potro of Argentina, who defeated Filip Krajinovic 6-4 6-2 for his 14th consecutive victory as the fifth seed looks to back up his Indian Wells title triumph with another tournament win.
"I will try to keep going and see how far I can go," said Del Potro, who next meets big-serving Milos Raonic after the Canadian cruised to a 6-3 6-4 victory against an overpowered Jeremy Chardy of France.
Del Potro beat Raonic in last week's semi-finals in California before edging out Federer in the title match.
"It'll be a difficult match," Raonic admitted. "He's been the best player in the world for the past four, five weeks." Elsewhere, Spaniard Pablo Carreno Busta crushed compatriot Fernando Verdasco 6-0 6-3 to reach the second Masters 1000 quarter-final of his career.
He will face South Africa's Kevin Anderson, who used his big serve to full effect in a 7-6(3) 6-4 win over 20-year-old American Frances Tiafoe.
(Reporting by Rory Carroll and Jahmal Corner; Editing by John O'Brien)
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