Sixth seed Shelton, a semi-finalist in New York two years ago, won 6-3, 6-2, 6-4 to stay on course for a potential quarter-final meeting with Carlos Alcaraz.
"Starting to feel like home for me," Shelton said after wrapping up victory in two hours, seven minutes on the main Arthur Ashe Stadium showcourt.
"This is the best court in tennis. It's my favourite place to play, favourite court, favourite tournament."
The big-serving 22-year-old is among a clutch of American men hoping to end the USA's 22-year wait for a Grand Slam singles title.
Shelton was less than a year old when Andy Roddick was the last American man to win a Slam with his victory at the 2003 US Open.
Shelton, though, insists he is not looking too far ahead following his opening win.
"One step at a time," Shelton said after completing his victory.
"Trying to get better every day," he added.
"As soon as you start looking ahead, you stumble over your own feet. So just one game at a time."
Shelton's preparations for the final Grand Slam of the year included his first ATP Masters title with a win at the Canadian Open earlier this month.
Shelton faces either Pablo Carreno Busta or Pablo Llamas Ruiz in the second round.
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