The 29-year-old defeated Frenchman Terence Atmane 6-3 7-6 (7-2) in 97 minutes, one of the eight matches on Monday (Tuesday AEST) decided in straight sets.
World No.3 Zverev will face Jakub MensÃk in the fourth round, after the Czech beat 13th seed Karen Khachanov 6-4 7-6 (13-11) and provided he is fit to play.
The German complained of physical problems after the match against Atmane, but declined to specify the issue.
Alongside Italian world No.1 Jannik Sinner, Zverev is the top title contender at the claycourt tournament, which carries a prize fund of around $US8.2 million ($A11.4 million). His chances have grown following the withdrawals of Carlos Alcaraz and Novak Djokovic.
Elsewhere, four more top-20 ranked players progressed with little fuss.
Russian Daniil Medvedev (10), Italy's Flavio Cobolli (13), Norwegian Casper Ruud (15) - the defending champion - and Francisco Cerandulo (20) of Argentina kept their title hopes alive.
Cerundolo's 6-2 6-3 win over Luciano Darderi was highlighted by a point after reaching over the net to get to a high return that spun back into the Italian's side of the court.
There was also an unusually long rally in the Ruud-Alejandro Davidovich Fokina match when the home favourite resorted to 15 straight lob shots to the back of the court.
The high returns in the 32-shot rally kept Rudd from attacking until one shot came up short and allowed him to power a forehand and take the point.
Daniel Merida's 6-4 6-2 loss later in the day to Greek Stefanos Tsitsipas means 19-year-old Rafael Jodar as the only Spaniard left in the tournament.
With AP & AAP.
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