The big-serving fourth seed blasted 15 aces on Wednesday and employed a drop-shot finesse at the net to preserve his hopes of capturing his first major after finishing runner-up at Roland Garros.
Zverev proved too much for the world No.77. Muller committed 41 unforced errors and battled through an apparent leg injury in brutal temperatures and high humidity at Flushing Meadows.
The German, beaten men's finalist in New York in 2020, came out firing, breaking Muller's serve in the first game before ultimately claiming the back-and-forth first set.
Clutching his upper right leg, Muller left the court for a lengthy medical time-out between sets but returned with a vengeance.
The resilient Frenchman stood two points from taking the second set but the world No.4 fought back to take the final five points of a tiebreaker.
Zverev cruised through the third set, closing out the match with a final ace.
Zverev will face Argentina's Tomas Etcheverry, who beat fellow countryman Francisco Cerundolo 6-3 4-6 6-4 1-6 6-3.
Big-hitting Frances Tiafoe did his bit to boost American hopes. Tiafoe was leading 6-4 6-1 1-0 when Kazakhstan's Alexander Shevchenko retired.
Tiafoe, who reached the New York semi-finals in 2022, said: "Happy to get it done. Hate to win it that way but overall I thought I actually played pretty well today."
It may have been a stress-free victory but Tiafoe still changed shirts five times.
Jiri Lehecka of the Czech Republic came back from two sets down to survive a stern test against American qualifier Mitchell Krueger 6-7 (7-5) 0-6 6-4 6-4 7-5.
Dutchman Tallon Griekspoor reached the third round for the first time in his career after 21st seed Sebastian Baez of Argentina withdrew from their match in the second set.
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