After having his tournament opener against Andrea Guerrieri postponed until day four, de Minaur had to endure another two-hour rain delay before putting away the Italian qualifier in straight sets.
The world No.7 advanced with a 6-2 6-2 6-4 victory to set up a second-round clash with Dutchman Botic van de Zandschulp.
In a clinical display, de Minaur broke Guerrieri twice in all three sets and only dropped serve once, when he was trying to close out the match at 5-2 in the third.
But New York's miserable weather has left de Minaur facing a scheduling backlog and needing to win seven best-of-five-set matches in 12 days to lift the trophy.
The three-times quarter-finalist will need to be back on site on Thursday (Friday AEST) to face the 70th-ranked van de Zandschulp and again the following day if he wins again.
Former champion Daniil Medvedev and American home hope Alex Michelsen were already safely through to round three before de Minaur even made it on court for his tournament opener.
When he finally opened his campaign on Wednesday (Thursday AEST), de Minaur looked intent on making up for lost time, racing to a 6-2 4-2 lead in just 49 minutes before the heavens opened.
After scampering back to the locker room, de Minaur returned to court 11 to wrap up victory in less than an hour.
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