Murray, 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.
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 faces Australian John Millman or American Emilio Nava in the second round.
