The 28-year-old German, who was dispatched in the second round last year, appeared to be heading for another early exit when Garin stepped up to serve for a 5-0 lead.
But the 29-year-old failed to press home his advantage as Zverev recouped a couple of games.
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At 5-3 up, Garin was unable to serve out the match and Zverev eventually levelled at 5-5.
He claimed Garin’s service to lead 6-5 with his own opportunity to serve for the match.
But Zverev wobbled.
Garin had two separate chances to push the match into a tiebreak before Zverev had a match point which he then squandered.
Another Zverev match point came and went before Garin had another chance to take the tie into the shoot-out.
But Zverev prevailed to set up a clash with the unseeded Belgian Zizou Bergs.
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“My level wasn’t there at all,” said Zverev after a tussle lasting two hours and 50 minutes.
“It’s the first clay-court match in 11 months and I didn’t have much time to prepare.
“I wasn’t feeling the ball so I didn’t feel like I could hit aggressive shots. But the most important thing is to get the win.”
Zverev and Alcaraz strive to be the new name on French Open men’s singles trophy
There were no such traumas for the former world number six Matteo Berrettini.
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He walloped the seventh seed Daniil Medvedev 6-0, 6-0 in 49 minutes.
The injury-plagued Italian was given an invitation to compete in the main draw by the tournament organisers and the 29-year-old advanced to the last-16 after condemning Medvedev to his first defeat on the senior tour in which he failed to win a game.
“I think I missed three shots for the match,” said Berrettini. “It’s not easy, especially with a tricky player like Daniil. The game plan was perfect, and my weapons were working.”
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Berrettini will take on the Brazilian Joao Fonseca who saw off the Frenchman Arthur Rinderknech 7-5, 4-6, 6-3.
Two-time French Open finalist Casper Ruud beat Corentin Moutet from France 7-5, 6-3.
The Norwegian will face Félix Auger-Aliassime from Canada after he dismissed the Croatian veteran Marin Cilic 7-6, 6-3.
