The German favourite looked at one point as if he might suffer the same fate as defending men's champion Andrey Rublev, who lost in three sets to Kazakh Alexander Bublik.
Zverev struggled early but recovered to fend off a gritty performance by Davidovich Fokina in their third-round clash and prevail 2-6 7-6 (7-3) 7-6 (7-0).
Zverev, seeking his third Madrid Open title, has won seven matches in a row and is coming off a title-run in Munich but he was slow out of the blocks on Sunday.
"The first set wasn't my best tennis but it is sport and it can change quickly," Zverev said.
"I was down a set and a break and I had to fight. I am very happy with the win. Alejandro is playing unbelievable tennis, the best of his life, so I am happy to be through."
The top seed will next face either Francisco Cerundolo or Francisco Comesana.
Rublev was eliminated in a 6-4 0-6 6-4 loss to Bublik, who had knocked out Australian Alexei Popyrin in the previous round.
It was the 10th top-10 win for the 75th-ranked Bublik, who made it to the fourth round in Madrid for the fourth time.
Rublev beat Felix Auger-Aliassime in last year's final in the Spanish capital.
Bublik will next face Ben Shelton or Miami champion Jakub Mensik in the round of 16.
Brandon Nakashima got past Flavio Cobolli to reach the fourth round of a Masters 1000 tournament for the fifth time. The 32nd-ranked American will next meet either Daniil Medvedev or Juan Manuel Cerundolo.
