The 35-year-old Scot, who has battled various injuries in recent years, raced to a 5-2 lead early in the match but dropped his level slightly before edging a tight first set and breaking quickly in the next to put 24th seed Cerundolo under pressure on a sweltering hot Arthur Ashe Stadium court.
“It is obviously nice any time, especially in those conditions and early on in a slam to get through in straight sets,” said Murray. “He’s a top player, it was a good win for me.
“Mentally it’s not easy going into a five-set match in those conditions after the way the last few weeks have been for me, and I did well.
“I did well physically and mentally to put that to the back of my mind and find a way to get through.”
Murray staved off a comeback attempt by Cerundolo to take the second set for a commanding lead and showed little signs of stepping off the gas pedal, as his opponent continued to make life difficult for himself with unforced errors.
A double break in the third set allowed twice Wimbledon champion Murray to surge ahead 4-1 and he sealed the win when Cerundolo sent a return long.
Murray next faces American wildcard Emilio Nava.