Murray, who has suffered badly with cramping during the hard-court season, earned a 7-5, 6-3, 6-3 win over 24th seed Cerundolo to advance to the second round.
The 35-year-old won a hard-fought opening set, with both players breaking serve in the first two games, and further break each took them to 5-5 before Murray won a crucial break point to take an early lead.
The Scot found his task much easier in the second set as he raced into a 5-0 lead, eventually taking it 6-3. Murray followed up with a dominant 5-1 lead. Cerundolo would break serve once more but it proved not to be enough as Murray served out to take the final set 6-3 and a place in the next round.
Murray’s appearance at the US Open this week marks a decade since he won his first Grand Slam at Flushing Meadows in 2012.
He has has two men’s singles titles at Wimbledon (2013 and 2016) and the London 2012 and Rio 2016 Olympic Games.