Zverev, ranked fifth in the world, recovered from a rusty opening set to break Ferrer’s service three times in the second set and twice more in the decisive stanza.
He will next face Australian Nick Kyrgios, who advanced with a 6-3 6-3 win over Fabio Fognini.
"I knew it was going to be tough. He has a lot of tricks up his sleeve and he's very unpredictable," Kyrgios said.
"We both competed. I just played a little better in some bigger points."
In earlier action, Fernando Verdasco dug deep to earn a 3-6 6-4 7-6(4) win over Australian qualifier Thanasi Kokkinakis to set up a fourth round clash against fellow Spaniard Pablo Carreno Busta.
The 34-year-old fired down 13 aces to beat Kokkinakis, who had been full of confidence after stunning world number one and defending champion Roger Federer in the second round.
Verdasco, who reached the Miami quarter-finals in 2009 and 2010, rallied from a 3-1 deficit in the third set and again from 3-0 down in the decisive tiebreaker before prevailing in a match that lasted nearly three hours.
He then got involved in a Twitter spat with Kyrgios, who had expressed his desire for the Spaniard to lose his third round match.
Next up for Verdasco is Carreno Busta, who beat American Steve Johnson 6-4 6-4 to reach the fourth round for the first time.
"It's never easy," Carreno Busta, the 16th seed, said after his 78-minute match.
"And this is the first time that I've played on this court. But I feel really comfortable and I'm playing good. When you play with confidence it's easier."
Also reaching the fourth round were Kevin Anderson, a 4-6 6-2 6-3 winner over Karen Khachanov, and Denis Shapovalov, who upset 11th-seeded American Sam Querrey 6-4 3-6 7-5.
Unseeded American Frances Tiafoe was leading his match against 10th seed Tomas Berdych, 6-7(2) 6-2 5-4 when it was suspended due to darkness.
(Reporting by Frank Pingue and Jahmal Corner, editing by Nick Mulvenney)
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